Graph and Election Problems Parameterized by Feedback Set Numbers

Because of the outdated latex system of arxiv, this is a less-featured v ersion of m y diploma theses. Please see http://robert.bredereck.info for the full-featured v ersion. F riedric h-Sc hiller-Univ ersit¨ at Jena D I P L O M A R B E I T Graph and Election Problems P a rameterized b y F eedback Set Numb ers zur Erlangung des ak ademischen Grades Diplom-Info rmatiker ausgef¨ uhrt am Lehrstuhl f¨ ur Theo retische Informatik I / Komplexit¨ atstheo rie an der F alkult¨ at f ¨ ur Mathematik und Info rmatik der F riedrich-Schiller-Universit¨ at Jena unter Anleitung von Dipl.-Bioinf. Nadja Betzler, Dipl.-Bioinf. Johannes Uhlmann und Prof. Dr. Rolf Niedermeier durch Rob ert Bredereck geb. am 31.01.1985 in W olgast Jena, 25. Mai 2010 Abstract This w ork in v estigates the parameterized complexity of three related graph mo diﬁcation problems. Giv en a directed graph, a distinguished vertex, and a p ositive integer k , Mini- mum Indegree Deletion asks for a v ertex subset of size at most k whose remov al makes the distinguished v ertex the only v ertex with minim um indegree. Minimum Degree Deletion is analogously deﬁned, but deals with undirected graphs. Bounded Degree Deletion is also deﬁned on undirected graphs, but has a p ositive in teger d instead of a distinguished v ertex as part of the input. It asks for a vertex subset of size at most k whose remo v al results in a graph in which ev ery vertex has degree at most d . The ﬁrst t wo problems hav e applications in computational so cial c hoice whereas the third problem is used in computational biology . W e inv estigate the parameterized complexit y with resp ect to the parameters “treewidth”, “size of a feedback v ertex set” and “size of a feedbac k edge set” resp ectively “size of a feedback arc set”. Each of these parameters measures the “degree of acyclicit y” in diﬀeren t w ays. F or Minimum Indegree Deletion we sho w that it is W[2]-hard with resp ect to b oth parameters that are deﬁned on acyclic graphs. W e describ e a branch-and-bound algorithm whose running time is O ( s · ( k + 1) s · n 2 ), where n is the num b er of vertices, k is the “num b er of vertices to delete”, and s is the “size of a feedbac k set”. F or Minimum Degree Deletion w e show W[1]-hardness with resp ect to the parameter “n umber of vertices to delete”. With resp ect to eac h of the parameters that measures the “degree of acyclicity” w e show ﬁxed-parameter tractability . W e describ e a simple searc h tree algorithm with running time O (2 s · n 3 ) where n is the n umber of v ertices and s is the “size of a feedback edge set” and tw o concrete ﬁxed-parameter algorithms with resp ect to the parameter “size of a feedback vertex set that do es not contain the distinguished vertex”. F or Bounded Degree Deletion we present a searc h-tree algo- rithm with running time O (3 s · n 2 ) where n is the num b er of vertices and s is the “size of a feedbac k edge set”. Zusammenfassung Diese Arb eit un tersuch t die parametrisierte Komplexit¨ at von drei v erwandten Graph- mo diﬁk ationsproblemen. Jedes dieser Probleme suc h t eine Knotenteilmenge b eschr¨ ank- ter Gr¨ oße, deren L¨ osc hung zu einem Graph mit einer problemsp eziﬁschen Eigensc haft f ¨ uhrt. Das erste b etrach tete Problem Minimum Indegree Deletion hat als Eingab e einen geric hteten Graphen, einen ausgewiesenen Knoten und eine nat ¨ urliche Zahl k . Die F rage ist, ob der ausgewiesene Knoten durc h L¨ osch ung v on maximal k Knoten der einzige Knoten mit minimalem Eingangsgrad werden k ann. Das Problem Minimum Indegree Deletion wurde im Kon text der W ahlforsc h ung eingef ¨ uhrt. Genauer gesagt analysiert man f ¨ ur W ahlsysteme ein ” Con trol “ -Szenario, in welc hem man fragt, ob durch L¨ osch ung einer durch die Gr¨ oße b eschr¨ ankten Kandidatenmenge das Ergebnis so zu b eeinﬂussen ist, dass ein ausgew¨ ahlter Kandidat gewinnt. Die Ergebnisse b ez ¨ uglic h Minimum In- degree Deletion lassen sic h eins zu eins auf dieses Szenario ¨ ub ertragen. Das zw eite b etrac htete Problem Minimum Degree Deletion ist Analog zu Minimum Indegree Deletion auf ungerich teten Graphen deﬁniert. Hier fordert man, dass nach L¨ osc hung der Knotenmenge ein ausgew¨ ahlter Knoten als einziger minimalen Grad hat. Auch dieses Problem l¨ asst sic h durc h ein nat ¨ urliches, auf sozialen Netzwerk en basierendes W ahlsystem motivieren. Das dritte b etrach tete Problem Bounded Degree Deletion ist eb enfalls auf ungeric hteten Graphen deﬁniert. Es hat jedoch im Gegensatz zu den beiden v orherigen Problemen keinen ausgewiesenen Knoten als T eil der Eingab e sondern fordert, dass nach L¨ osch ung v on maximal k Knoten alle Knoten maximal Grad d b esitzen f¨ ur ein b estimm tes zur Eingab e geh¨ origes d . Das Problem Bounded Degree Deletion ﬁndet Anw endung b ei der Analyse v on biologischen Netzwerk en. Neb en ¨ Ahnlic hkeiten in der Deﬁnition hab en die drei b etrach teten Probleme gemeinsam, dass sie auf kreisfreien Graphen in Polynomzeit l¨ osbar sind, w¨ ahrend sie jedoch im All- gemeinen NP-sc hw er sind. Diese Eigenschaft war Ausgangspunkt f¨ ur die Un tersuc hung der parametrisierten Komplexit¨ at b ez ¨ uglic h drei v ersc hiedener P arameter, w elc he jew eils auf un te rsc hiedliche W eise die Distanz des Eingab egraphen zu einem kreisfreien Graphen messen. Der erste dieser Parameter ist die f ¨ ur die Untersuc h ung ungerich teter Graphen w eit v erbreitete ” Baum weite “ . Die b eiden anderen Parameter messen jew eils die Anzahl der Knoten bzw. Kanten deren L¨ osch ung zu einem kreisfreien Graphen f ¨ uhrt. Genauer gesagt sind dies die Parameter ” Gr¨ oße einer kreiskritisc hen Knotenmenge “ und ” Gr¨ oße einer kreiskritisc hen Kantenmenge “ . F ¨ ur Minimum Indegree Deletion zeigen wir, dass es b ez ¨ uglic h b eider auf gerich- teten Graphen deﬁnierten P arameter W[2]-sc hw er und damit oﬀenbar nich t festparam- eterhandhabbar ist. Betrac htet man hingegen den P arameter ” Anzahl zu l¨ oschender Knoten “ in Kombination mit einem der b eiden anderen, l¨ asst sich F estparameterhand- habbark eit nach w eisen. Wir b eschreiben einen einfachen Algorithmus dessen Laufzeit O ( s · ( k + 1) s · n 2 ) ist, w ob ei n die Anzahl der Knoten, k die ” Gr¨ oße der zu l¨ osc henden Knotenmenge “ und s die ” Gr¨ oße einer kreiskritischen Menge “ ist. F ¨ ur Minimum Degree Deletion wird zus¨ atzlic h der Parameter ” Anzahl zu l¨ osc hender Knoten “ b etrach tet und f ¨ ur diesen W[1]-Sc h were nac hgewiesen. Bez¨ uglich jedem der Parameter ” Baum weite “ , ” Gr¨ oße einer kreiskritischen Knotenmenge “ so wie ” Gr¨ oße einer kreiskritischen Kanten- menge “ wird F estparameterhandhabbarkeit gezeigt. Wir zeigen b ez ¨ uglich ” Gr¨ oße einer kreiskritisc hen Kantenmenge “ einen Problemkern, dessen Gr¨ oße linear in der Anzahl der Knoten ist. Im Gegensatz dazu zeigen wir b ez ¨ uglic h der anderen b eiden P arameter, dass es keinen Problemk ern p olynomieller Gr¨ oße gibt, es sei denn, die Polynomialzeithierar- 5 c hie kollabiert. Wir b eschreiben einen Such baumalgorithm us mit Laufzeit O (2 s · n 3 ), w ob ei n die Anzahl der Knoten und s die ” Gr¨ oßes einer kreiskritischen Kan tenmenge “ ist. Zw ei konkrete F estparameteralgorithmen werden b ez ¨ uglic h des Parameters ” Gr¨ oße einer kreiskritisc hen Knotenmenge, welc he nich t den ausgewiesenen Knoten enth¨ alt “ pr¨ asen- tiert. Wir w eisen absc hließend f ¨ ur Bounded Degree Deletion F estparameterhandhab- bark eit b ez ¨ uglich des P arameters ” Gr¨ oße einer kreiskritisc hen Kan tenmenge “ nach. Ein en tsprechender Algorithmus hat die Laufzeit O (3 s · n 2 ), wobei n die Anzahl der Knoten und s die ” Gr¨ oßes einer kreiskritischen Kantenmenge “ ist. Contents 1 Introduction 9 1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2 P arameterized complexity . . . . . . . . . . . . . . . . . . . . . . . . 12 2 Degree-based vertex deletion p roblems 17 2.1 Minim um Indegree Deletion . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Minim um Degree Deletion . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Bounded Degree Deletion . . . . . . . . . . . . . . . . . . . . . . . . 19 3 F eedback sets 21 4 Minimum Indegree Deletion 25 4.1 Kno wn results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2 F eedbac k vertex/arc set size as parameter . . . . . . . . . . . . . . . 26 4.3 F eedbac k vertex set size and solution size as combined parameter . . 31 5 Minimum Degree Deletion 35 5.1 Solution size as parameter . . . . . . . . . . . . . . . . . . . . . . . . 36 5.2 Size of a feedbac k edge set as parameter . . . . . . . . . . . . . . . . 39 5.3 T reewidth as parameter . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.3.1 Monadic second-order logic . . . . . . . . . . . . . . . . . . . . 43 5.3.2 MSO expression for Minim um Degree Deletion . . . . . . . . . 45 5.4 Size of a feedbac k vertex set as parameter . . . . . . . . . . . . . . . 47 5.4.1 In teger linear programming . . . . . . . . . . . . . . . . . . . 49 5.4.2 Dynamic programming . . . . . . . . . . . . . . . . . . . . . . 53 5.5 No p olynomial k ernel with resp ect to s v . . . . . . . . . . . . . . . . 57 6 Bounded Degree Deletion 63 6.1 Kno wn results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 6.2 Size of a feedbac k edge set as parameter . . . . . . . . . . . . . . . . 63 7 Conclusion and Outlo ok 69 1 Intro duction This work inv estigates the parameterized complexity of three related graph mo diﬁ- cation problems. Eac h of these problems asks for a v ertex subset of b ounded size whose remo v al results in a graph that satisﬁes a problem-sp eciﬁc prop erty . The ﬁrst considered problem Minimum Indegree Deletion has a directed graph, a distinguished v ertex, and a p ositive integer k as input. The question is whether the distinguished v ertex can b e made the only v ertex with minimum indegree by remo ving at most k v ertices. The problem Minimum Indegree Deletion w as in- tro duced in the con text of computational so cial choice [BU09]. More precisely , one analyzes a so-called “control” scenario for a voting rule in whic h one asks to ma- nipulate the result of the voting rule by remo ving a size-b ounded candidate subset suc h that a distinguished candidate b ecomes the only winner. The results regarding Minimum Indegree Deletion can b e transferred to this scenario [BU09]. The second considered problem Minimum Degree Deletion is deﬁned analogously to Minimum Indegree Deletion , dealing with undirected graphs. Here one asks whether it is p ossible to make a distinguished vertex the only v ertex with mini- m um degree by removing a subset of vertices. This problem can b e motiv ated b y a simple v oting rule that is based on a social netw ork. The third considered prob- lem Bounded Degree Deletion [Mos10] is also deﬁned on undirected graphs, but in contrast to both previous problems it has no distinguished v ertex as part of the input. It asks for a set of k vertices whose remov al results in a graph with maxim um degree d for a sp eciﬁc p ositive integer d that is part of the input. The problem Bounded Degree Deletion has applications in the analysis of genetic net works [Mos10]. In Chapter 2 w e provide formal deﬁnitions and more details on applications. Besides similarities in the deﬁnition, all three problems hav e in common that they are solv able in polynomial time on acyclic graphs, but NP-hard in general. This prop ert y is the starting p oin t for the in v estigation of the parameterized complexit y with resp ect to three distinct parameters which measure the distance of the input graph to an acyclic graph in three diﬀeren t w a ys. The ﬁrst of these parameters is the “treewidth”, b eing well-kno wn in the analysis of undirected graphs. The tw o other parameters measure in eac h case the num b er of v ertices resp ectively edges whose remov al results in an acyclic graph. More precisely , w e hav e the parameters “size of a feedbac k v ertex set” and “size of a feedback edge set” resp ectiv ely “size of a feedback arc set” in the directed case. A formal deﬁnition of “treewidth” is given in Section 1.1 whereas the other parameters are considered in detail in Chapter 3. In what follo ws, w e brieﬂy describ e our results, also see Figure 1.1. F or Minimum Indegree Deletion we show that it is W[2]-hard, hence, that it is presumably not ﬁxed-parameter tractable with resp ect to b oth parameters that are deﬁned on acyclic graphs (see Section 4.2). Considering the parameter “num b er of vertices to delete” 10 parameter MID MDD BDD t w FPT op en s v W[2]-hard, in XP FPT op en s a/e W[2]-hard, in XP FPT FPT k W[2]-complete W[1]-hard, in XP W[2]-complete d FPT FPT FPT ( s v , k ) FPT FPT op en parameter description t w treewidth of the input graph s v size of a feedbac k v ertex set s a/e size of a feedbac k arc/edge set k size of a solution set d maxim um degree of a vertex Figure 1.1: Ov erview of the parameterized complexit y of Minimum Inde- gree Deletion (MID), Minimum Degree Deletion (MDD) and Bounded Degree Deletion (BDD). New results are in b oldface. The remaining results are obtained from [BU09] and [Mos10]. in combination with one of b oth parameters, w e show ﬁxed-parameter tractability . W e describ e a branch-and-bound algorithm whose running time is O ( s · ( k + 1) s · n 2 ), where n is the num b er of v ertices, k is the “num b er of vertices to delete”, and s is the “size of a feedback set” (see Section 4.3). F or Minimum Degree Deletion w e additionally consider the single parameter “num b er of vertices to delete” and sho w W[1]-hardness (see Section 5.1). With resp ect to each of the parameters “treewidth”, “size of a feedbac k vertex set”, and “size of a feedback edge set” we sho w ﬁxed-parameter tractabilit y (see Section 5.3.2). W e show with resp ect to “size of a feedback edge set” a problem k ernel whose size is linear in the n umber of vertices (see Section 5.2). In contrast, we show that with resp ect to each of the t w o other parameters there is no problem kernel of p olynomial size, assuming that the p olynomial-time hierarch y do es not collapse (see Section 5.5). W e describ e a simple search tree algorithm with running time O (2 s · n 3 ) where n is the n umber of v ertices and s is the “size of a feedback edge set”. This algorithm complemen ts the kernelization result from Section 5.2. W e present tw o concrete ﬁxed-parameter algorithms with resp ect to the parameter “size of a feedback vertex set that do es not con tain the distinguished vertex”. The ﬁrst one uses the tec hnique of in teger linear programming (see Section 5.4.1) whereas the second one is a dynamic programming algorithm (see Section 5.4.2). Besides the intractabilit y results in Section 4.2 and Section 5.1, the tw o algorithms with resp ect to the parameter “size of a feedbac k v ertex set that do es not con tain the distinguished vertex” are the most demanding parts of this work. Finally , we sho w ﬁxed-parameter tractabilit y for Bounded Degree Deletion with resp ect to “size of a feedback edge set”. A corresp onding searc h-tree algorithm has running time O (3 s · n 2 ) where n is the num b er of vertices and s is the “size of a feedbac k edge set”. 1 Introduction 11 Organization of this wo rk. The second c hapter giv es an ov erview of the consid- ered problems and its applications. A third chapter con tains some deﬁnitions and computational asp ects of the considered parameters measuring the distance of the input graph from an acyclic graph. Each of the Chapters 4-6 in v estigates one of the considered vertex deletion problem. The last c hapter pro vides a conclusion and outlo oks to further in vestigations in this ﬁeld. 1.1 Prelimina ries W e in tro duce the basic terms and metho ds whic h are necessary in this work. Graph theo ry . All computational problems that are inv estigated in this work are graph problems. An undir e cte d gr aph G is a pair ( V , E ), where V is a ﬁnite set of vertic es and E is a ﬁnite set of e dges which are unordered pairs of vertices. A dir e cte d gr aph G is also a pair ( V , A ), where V is a ﬁnite set of v ertices, but A is a ﬁnite set of ar cs whic h are ordered pairs of v ertices. All graphs considered in this w ork are simple , that is, there are no m ulti-arc/multi-edges, and they do not con tain self-lo ops , that is, no edge/arc from a vertex to itself. Let { v 1 , v 2 } b e an edge of an undirected graph. Then, v 1 is a neighb or of v 2 and vice versa. W e denote deg ( x ) as the de gr e e of the v ertex x , that is, the num b er of its neighbors. F urthermore, the (op en) neighb orho o d of a vertex v in an undirected graph G = ( V , E ) is deﬁned as N ( v ) := { u | { u, v } ∈ E } . Let ( v 1 , v 2 ) b e an arc of a directed graph. W e denote v 1 as inneighb or of v 2 and v 2 as outneighb or of v 1 . The inde gr e e of a vertex is the n um b er of its inneigh b ors and the outde gr e e of a v ertex is the n umber of its outneigh b ors. The (op en) inneighb orho o d of a v ertex v in a directed graph G = ( V , A ) is deﬁned as N in ( v ) := { u | ( u, v ) ∈ A } and the (op en) outneighb orho o d of a vertex v in a directed graph G = ( V , A ) is deﬁned as N out ( v ) := { u | ( v , u ) ∈ A } . F or a v ertex set S ⊆ V , we write G [ S ] to denote the graph induced b y S in G , that is, G [ S ] := ( S , { e ∈ E | e ⊆ S } ) for an undirected graph G = ( V , E ) resp ectiv ely G [ S ] := ( S, { ( x, y ) ∈ A | x ∈ S ∧ y ∈ S } ) for a directed graph G = ( V , A ). F or a subset S ⊆ V , w e also write G − S instead of G [ V \ S ]. W e deﬁne P i := ( V P i := { v 1 , . . . , v i } , E P i := {{ a, b } | 1 ≤ a < b ≤ i } ) as p ath of length i b et ween v 1 and v i . Moreo ver, C i := ( V P i , E P i ∪ { v i , v 1 } ) is deﬁned as cycle of length i . A graph that con tains a cycle as subgraph is called cyclic . Otherwise we say the graph is acyclic . In directed graphs the terms p ath of length i ( P i := ( V P i := { v 1 , . . . , v i } , E P i := { ( a, b ) | 1 ≤ a < b ≤ i } )), cycle of length i ( C i := ( V P i , E P i ∪ ( v i , v 1 ))), and cyclic/acyclic are deﬁned analogously . An undirected graph is c onne cte d if there is a path b et ween each tw o v ertices. A connected and acyclic graph is called a tr e e . T ree decomp osition and treewidth. As mentioned in the in tro duction, man y hard graph problems are easy when restricted on acyclic graph. The main question is: “Wh y is an NP-hard problem in P when restricted on a tree?”. W e need a measure of the “tree-likeness” of a given graph. Robertson and Seymour [RS86] introduced the concept of tree-decomp ositions and treewidth of undirected graphs. 12 Deﬁnition 1. L et G = ( V , E ) b e an undir e cte d gr aph. A tr e e de c omp osition of G is a p air ( { X i | i ∈ I } , T ) wher e e ach X i is a subset of V , c al le d a b ag , and T is a tr e e with the elements of I as no des. The fol lowing thr e e pr op erties must hold: 1. S i ∈ I X i = V , 2. for every e dge { u, v } ∈ E , ther e is an i ∈ I such that { u, v } ⊆ X i , and 3. for al l i, j, k ∈ I , if j lies on the p ath b etwe en i and k in T , then X i ∩ X k ⊆ X j . The width of the tr e e de c omp osition ( { X i | i ∈ I } , T ) e quals max {| X i | | i ∈ I } − 1 . The tr e ewidth of G is the minimum k such that G has a tr e e de c omp osition of width k . Clearly , a (non-trivial) tree has treewidth 1, a cycle has treewidth 2, and a complete graph with n v ertices has treewidth n − 1. The other t wo parameters whic h measure the “degree of acyclicit y” are introduced in Chapter 3. 1.2 P a rameterized complexit y Man y interesting problems in computer science are computationally hard problems in w orst case. The most famous class of such hard problems is the class of NP-hard problems. The relation b etw een P (whic h includes the “eﬃcien t solv able problems”) and NP is not completely clear at the moment. Even if P = NP it is not self-evident that we are able to design eﬃcient polynomial-time algorithms for eac h NP-hard problem. But we hav e to solv e NP-hard problems in practice. Thus, according to the state of the art of computational complexit y theory , NP-hardness means that w e only hav e algorithms with exp onen tial running times to solv e the corresp onding problems exactly . This is a h uge barrier for practical applications. There are dif- feren t w ays to cop e with this situation: heuristic metho ds, randomized algorithms, a verage-case analysis (instead of w orst-case) and approximation algorithms. Unfor- tunately , none of these metho ds provides an algorithm that eﬃciently computes an optimal solution in the worst case. Since there are situations where we need p erfor- mance and optimality guarantee at least for a sp eciﬁed type of input, another w a y out is needed. Fixed-parameter algorithms pro vide a p ossibility to redeﬁne problems with several input parameters. The main idea is to analyze the input structure to ﬁnd parameters that are “resp onsible for the exp onential running time”. The aim is to ﬁnd suc h a parameter, whose v alues are constan t or “logarithmic in the input size” or “usually small enough” in the problem instances of your application. Thus, w e can sa y something like “if the parameter is small, w e can solv e our problem in- stances eﬃcien tly”. W e will use the tw o-dimensional parameterized complexit y theory [DF99, Nie06, F G06] for studying the computational complexity of sev eral graph problems. A p ar ameterize d pr oblem (or language) L is a subset L ⊆ Σ ∗ × N for some ﬁnite al- phab et Σ. F or an elemen t ( x, k ) of L , b y conv en tion x is called pr oblem instanc e 1 1 Most parameterized problems originate from classical complexity problems. One can interpret x as the input of the original/non-parameterized problem. 1 Introduction 13 and k is the p ar ameter . The tw o dimensions of parameterized complexit y theory are the size of the input n := | ( x, k ) | and the parameter v alue k , whic h is usually a non-negativ e integer. A parameterized language is called ﬁxe d-p ar ameter tr actable if we can determine in f ( k ) · n O (1) time whether ( x, k ) is an element of our language, where f is a computable function only dep ending on the parameter k . The class of ﬁxed-parameter tractable problems is called FPT. Th us, it is very important to ﬁnd go o d parameters. In the follo wing c hapters, w e need four of the core to ols in the dev elopment of parameterized algorithms [Nie06]: data reduction rules (ker- nelization), (depth-b ounded) searc h trees, dynamic programming and integer linear programs. The idea of kernelization is to transform an y problem instance x with parameter k in p olynomial time into a new instance x 0 with parameter k 0 suc h that the size of x 0 is b ounded from ab o ve by some function only dep ending on k and k 0 ≤ k , and ( x, k ) ∈ L if and only if ( x 0 , k 0 ) ∈ L . The reduced instance ( x 0 , k 0 ) is called pr oblem kernel . This is done by data r e duction rules , which are transformations from one problem instance to another. A data reduction rule that transforms ( x, k ) to ( x 0 , k 0 ) is called sound if ( x, k ) ∈ L if and only if ( x 0 , k 0 ) ∈ L . Besides kernelization we use (depth-b ounded) se ar ch tr e es algorithms . A search algorithm tak es a problem as input and returns a solution to the problem after ev aluating a num b er of p ossible solutions. The set of all p ossible solutions is called the search space. Depth-b ounded search tree algorithms organize the systematic and exhaustiv e exploration of the search place in a tree-lik e manner. Let ( x, k ) denote the instance of a parameterized problem. The search tree algorithm replaces ( x, k ) b y a set H of smaller instances ( x i , k i ) with | x i | < | x | and k i < k for 1 ≤ i ≤ | H | . Th us, the search tree size (n umber of no des) is clearly b ounded by | H | k . Since the running time of the replacement pro cedure is b ounded b y a p olynomial in the instance size, a constan t-size set H alw ays leads to a ﬁxed-parameter algorithm with resp ect to k . How ev er, there are more reﬁned metho ds to compute a better upp er b ound for the search tree size using the so-called branching v ector, but they are not imp ortan t in this work. Another imp ortan t technique used in this work is dynamic pr o gr amming . It go es bac k to Bell [Bel03]. As well as search trees dynamic programming makes exhaus- tiv e search, but is more eﬃcien t by av oiding the computation of subproblems more than once. It is used when a problem exhibits the prop erty of having optimal sub- structur e , that is, an optimal solution to the problem contains within it optimal solutions to subproblems. Two further prop erties are imp ortant for a feasible dy- namic programming: Indep endenc e , that is, the solution of one subproblem do es not aﬀect the solution of another subproblem of the same problem, and overlapping subpr oblems , that is, the same problem o ccurs as a subproblem of diﬀeren t problems. One organizes the computation of the solutions for the subproblems in the dynamic pr o gr amming table . An established w ay to compute the table which is used in this w ork is the b ottom-up computation. Typically in ﬁxed-parameter algorithms, the computation of a single table entry dep ends only on a constant n umber of parent en tries and can b e done in p olynomial time. The table size is b ounded by a func- tion only dep ending on the parameter. F urthermore, the o v erall-solution can b e computed from the completely ﬁlled table in p olynomial time. 14 The technique of inte ger line ar pr o gr amming is a sp ecial case of line ar pr o gr amming whic h go es bac k to Kan torovic h [Kan60]. It is a tec hnique for the optimization (minimization and maximization) of a (linear) obje ctive function , sub ject to linear equalit y and linear inequality contains. In an inte ger line ar pr o gr am (ILP), the unkno wn v ariables are required to b e integers instead of real n umber as in a linear program. Whereas solving a linear program is p ossible in p olynomial time, integer linear programs are NP-hard [Kar72]. Anyho w, the technique can b e used to show ﬁxed-parameter tractabilit y when the num b er of unkno wn v ariables is b ounded by a function only dep ending on the parameter [Len83, Kan87]. More ab out these four tec hniques can b e found in [Nie06]. In many applications one is in terested in deciding an NP-hard problem or computing the optimal solution of the corresp onding search or optimization problem. There- fore, ﬁxed-parameter tractability is a desired attribute of a problem together with a w ell-chosen parameter. How ev er, there are also some cases where intractabilit y can also b e a desired attribute. F or example voting rules seem to b e “more fair” if a (at this p oint not more speciﬁed) manipulation or control of that rule is computationally hard. Th us, parameterized intractabilit y can b e a p ositiv e result as well as negative result. In this work, we use a characterization of parameterized problems that pro- vides ev en more than only determining ﬁxed-parameter tractabilit y or in tractability . In analogy to the concepts of NP-hardness, NP-completeness, and p olynomial-time man y-to-one reductions in classical complexity theory , Do wney and F ellows [DF99] dev elop ed a framew ork of reductions and a hierarch y of parameterized complexit y classes. Deﬁnition 2. A p ar ameterize d r e duction fr om a p ar ameterize d pr oblem L ⊆ Σ ∗ × N to another p ar ameterize d pr oblem L 0 ⊆ Σ ∗ × N is a function that, given an instanc e ( I , k ) , r eturns in time f ( k ) · p oly ( | ( I , k ) | ) an instanc e ( I 0 , k 0 ) , with k 0 only dep ending on k , such that ( I , k ) ∈ L if and only if ( I 0 , k 0 ) ∈ L 0 . A parameterized problem L b elongs to W[ t ] if L can b e reduced to a weigh ted sat- isﬁabilit y problem for the family of circuits of depth at most some function only dep ending on the parameter and weft at most t , where w eft is the maximum n um- b er of gates with unrestricted fan-in on an input-output path in the circuit. Sim- ilar to the classical complexit y theory w e denote a problem as W[ t ]-hard if there is a parameterized reduction from a problem that is already known to b e W[t]- complete. In this work, the most imp ortant thing we ha v e to know ab out the W[ t ]-hierarc hy is that ev ery parameterized problem which is W[ t ]-hard for t ≥ 1 is b elieved to b e not ﬁxed-parameter tractable. (This holds under the separation h yp othesis FPT 6 =W[1].) A parameterized problem L b elongs to the class XP if it can b e determined in f ( k ) · | x | g ( k ) time whether ( x, k ) ∈ L where f and g are computable functions only dep ending on the parameter k . It holds that FPT ⊆ W[1] ⊆ W[2] ⊆ .. ⊆ XP . More details ab out this ﬁelds can b e found in [DF99, FG06, Nie06]. An established w ay of proving W[t]-hardness is to giv e a parameterized reduction from a problem that is already known to b e W[t]-hard. In this work, we use the follo wing tw o problems: 1 Introduction 15 Independent Set Given: An undirected graph G = ( V , E ) and an integer k ≥ 1. Question: Is there a subset V 0 ⊆ V of size at least k suc h that there are no edges in G [ V 0 ]? Domina ting Set Given: An undirected graph G = ( V , E ) and an integer k ≥ 1. Question: Is there a subset V 0 ⊆ V of size at most k suc h that ev ery v ertex in V either b elongs to V 0 or has a neigh b or in V 0 ? The Independent Set problem with resp ect to the parameter k is known to b e W[1]-complete and the Domina ting Set problem with resp ect to the parameter k is kno wn to b e W[2]-complete [DF99]. 2 Degree-based vertex deletion p roblems In this w ork, we analyze three quite similar graph problems. All problems get a directed or undirected graph as one part of the input. In each problem one asks to delete a subset of v ertices of bounded size to get a mo diﬁed graph that satisﬁes a problem-sp eciﬁc prop erty . F urthermore, this prop ert y dep ends in eac h of the three problems on the degree of the vertices. Ho w ever, the particular problems hav e quite diﬀeren t applications. In the follo wing three sections, w e give a formal deﬁnition and a motiv ation (“Why is this problem relev an t?”). 2.1 Minimum Indegree Deletion Recen tly , so cial choice problems b ecame imp ortant in the ﬁelds of computational complexit y and algorithmics. In this con text, the in v estigation of v oting systems is a relev an t area. There are t wo recent surv eys by Chev aleyre et al. [CELM07] and F al- iszewki et al. [FHHR09b]. The most obvious application of v oting systems might b e p olitical elections. There are also sev eral applications in the ﬁelds of rank aggrega- tion and multi-agen t systems. Besides w ork that fo cuses on the problem to determine the winner or an optimal ranking for diﬀeren t voting systems, a signiﬁcant num b er of pap ers also inv estigates how an external agen t or a group of voters can inﬂuence the election in fav or or disfa vor of a distinguished candidate. Concrete scenarios of inﬂuencing are manipulation [BTT89, BFH + 08, CSL07, MPRZ08], electoral con- trol [BTT92, FHHR09a, HHR05], lobb ying [CFRS07], and brib ery [FHHR09a]. This sho ws that in vestigations in this ﬁeld are of high interest. In this section we present a directed graph problem that is closely related to control in the Llull voting rule. W e start with an introduction to this rule and in tro duce the corresponding graph problem. Llull voting. F ormally , an ele ction ( V , C ) consists of a m ultiset of votes V and a set of candidates C . A vote is a preference list ov er all candidates. In an election, w e either ask for a winner , that is, one of the candidates who are “b est” in the election, or for a unique winner . Of course, a unique winner do es not alwa ys exist. W e only consider the unique-winner case for our con trol v ariant, but our results can b e easily mo diﬁed to w ork for the winner case as w ell. The term Llull 1 v oting w as introduced by F aliszewski et al. [FHHR09a]. It is based on pairwise comparisons b etw een candidates: A candidate wins the pairwise contest against another candidate if it b eats the other candidate in more than half of the 1 Llull is the sp ecial case of Copeland α where α = 1. 18 v otes. The winner of a pairwise con test gets one p oint and the loser receiv es no p oin t. If tw o candidates are tied, b oth candidates get one p oint. A Llull winner is a candidate with the highest score. It is used for example in sp ort tournamen ts, chess, or in fo otball leagues, where the teams or pla y ers can b e considered as candidates. In the following w e consider the concept “con trol of a voting rule”. T o c ontr ol an election, an external or in ternal agen t, traditionally called the chair , is allo w ed or able to change the voting pro cedure to reach certain goals. F or example, a typical question is how man y candidates the c hair has to delete to make his/her fa vorite candidate a unique winner. In most cases it is a desirable attribute of a v oting rule to b e either immune to control, that is, it is imp ossible to con trol the voting rule, or at least to be resistan t to control, that is, the corresp onding decision problem is NP-hard [BTT92]. Unfortunately , NP-hardness do es only imply computational hardness in the worst case. There might b e sp ecial inputs where is it easy to decide the corresp onding problem. Regarding the complexity of con trol, Llull voting is resistant to constructive can- didate control and vulnerable for destructiv e candidate control [FHHR09a]. Thus, in vestigating the parameterized complexity of control of Llull v oting helps us to extend our knowledge ab out the “danger of con trol of Llull election”. This is what w e do in Chapter 4. Therefore, w e in tro duce the directed graph problem whic h corresp onds to candidate con trol in Llull elections. Minimum Indegree Deletion. A Llull election can be depicted by a directed graph where the candidates are represen ted as v ertices and there is an arc from vertex c to v ertex d if and only if the corresp onding candidate c defeats the corresp onding candidate d in the pairwise comparison con test. Ob viously , the Llull score of a candidate c can be considered as the total num b er of candidates min us the n um b er of candidates that beat c in the pairwise comparison. Th us, a Llull winner corresp onds to a v ertex with minim um indegree. Of course, the deletion of a vertex corresp onds to the deletion of a candidate in the election. These observ ations motiv ate the in tro duction of the follo wing directed graph problem which w as originally introduced in [BU09]: Minimum Indegree Deletion Given: A directed graph D = ( W , A ), a distinguished v ertex w c ∈ W , and an in teger k ≥ 1. Question: Is there a subset W 0 ⊆ W \ { w c } of size at most k suc h that w c is the only v ertex that has minimum indegree in D − W 0 ? The equiv alence of Minimum Indegree Deletion to constructiv e control b y delet- ing candidates in Llull elections w as shown in [BU09]. 2.2 Minimum Degree Deletion Constructiv e con trol b y deleting candidates in Llull elections leads to the directed Minimum Indegree Deletion . F rom the theoretical p oint of view, it is an in- tuitiv e task to inv estigate its undirected v arian t. F rom the practical p oint of view, 2 Degree-based vertex deletion problems 19 a corresp onding voting problem can b e formulated, to o: Giv en is a so cial network , that is, an undirected graph G = ( V , E ) where v ertices corresp ond to sub jects and edges corresp ond to relations b et w een tw o sub jects. F or example, consider the rela- tion “disharmon y b etw een t wo sub jects” and the sub jects are candidates of a voting rule where the candidate with fewest disharmonies wins. No w, the chair is ask ed to con trol the election by removing a sp eciﬁed num b er of candidates from the netw ork to mak e a sp eciﬁc candidate the winner of the election. The corresp onding graph problem Minimum Degree Deletion is given in the following: Minimum Degree Deletion Given: An undirected graph G = ( V , E ), a distinguished vertex w c ∈ V , and an in teger k ≥ 1. Question: Is there a subset V 0 ⊆ V \ { w c } of size at most k such that w c is the only v ertex that has minimum degree in G − V 0 ? Clearly , removing candidates from the so cial net work corresp onds to removing can- didates in the graph G . The equiv alence of the graph problem and constructiv e con trol of the voting rule by removing candidates is easy to see. 2.3 Bounded Degree Deletion The ﬁrst tw o problems ask for a vertex subset of a sp eciﬁc size whose remov al satisﬁes the graph prop ert y “only a distinguished vertex has minimum (in)degree”. The problem which is introduced in this section no-longer has a distinguished candidate, but an upp er b ound on the degree. This means, the underla ying graph prop erty is “the maxim um degree of the vertices in the graph is b ounded b y d ” for a speciﬁc p ositiv e integer d which is given in the input. Bounded Degree Deletion w as in tro duced in [Mos10] and is formally deﬁned as follows: Bounded Degree Deletion Given: An undirected graph G = ( V , E ), and integers d ≥ 0 and k ≥ 0. Question: Do es there exists a subset V 0 ⊆ V of size at most k whose remo v al from G yields a graph in which each v ertex has degree at most d ? Whereas Minimum Indegree Deletion and Minimum Degree Deletion cor- resp ond to problems in computational so cial choice, Bounded Degree Deletion has applications in computational biology: In the analysis of genetic netw orks based on micro arra y data, a cen tral to ol is to ﬁnd cliques or “near-cliques” [BCK + 05, CLS + 05]. Searching for cliques is a computational hard problem. Here, Ver tex Co ver as complemen tary dual problem to clique detection could b e used very suc- cessful instead of direct clique detection. A mathematical concept for near-cliques is the concept of s -plexes, that are, subsets of vertices such that eac h s -plex vertex is connected to all other v ertices in the s -plex but to s − 1. Note that cliques are 1- plexes. Bounded Degree Deletion is the complementary dual problem to s -plex detection [Mos10]. Since Bounded Degree Deletion is a vertex deletion prob- lem for the hereditary graph prop ert y “eac h vertex has degree at most d ” [Mos10], 20 NP-completeness is giv en due to a general result by Lewis et al. [L Y80]. It follo ws from the reduction in [L Y80] that Bounded Degree Deletion cannot be appro x- imated b etter than Ver tex Cover . Th us, an approximation lo wer b ound of 1 . 36 assuming P 6 = NP [DS05] as w ell as the APX-hardness of Ver tex Cover [PY91] can b e carried o ver. The b est known appro ximation factors are 2 [F uj98] for d = 1 and 2 + log ( d ) [OB03] for d ≥ 2. Finally w e in tro duce some general terms which are used in the analysis of all three problems. A yes-instance of each problem can b e detected through ﬁnding a vertex subset of b ounded size whose remov al satisﬁes the problem-sp eciﬁc graph prop erty . W e denote such subsets as solution sets . The result of remo ving a solution set from the input graph is called solution gr aph . 3 F eedback sets As mentioned in the in tro duction, parameterized algorithms are designed to conﬁne the combinatorial explosion to a parameter of the input instance. P opular param- eters are, for example, the solution size for problems asking for a sp eciﬁc “solution set”. Besides v ery natural parameters suc h as “n um b er of edges” in graph problems, “n umber of candidates” in v oting problems or “size of the alphab et” in string-based problems, one is often interested in structural parameters lik e “av erage distance” or “size of a v ertex cov er” (see [Nie10] for a survey). There is a huge class of graph problems whic h are easy to solve on acyclic graphs, but NP-hard in general. Later on, we will see that ev ery graph problem based on a graph property expressible b y some sp eciﬁc (non-trivial) logic is solv able in linear time on trees. The algorithms whic h follo w directly from this result are quite impractical, but it is a go o d classiﬁcation to ol. In most cases, there are simple and direct p olynomial-time algorithms for problems whose input graph is acyclic. In this wa y , a parameter that measures the “degree of acyclicity” of a graph is ma y b e a go o d candidate for ﬁxed-parameter tractability results. Often, one uses the treewidth and designs algorithms that op erate on tree decomp ositions. In this w ork, we will also fo cus on t wo weak er parameters which will b e introduced in this section. There are three reasons for working also on other parameters that measure the “degree of acyclicit y”, diﬀeren t from treewidth: The ﬁrst reason is that comput- ing an optimal tree decomp osition (or even the treewidth) is diﬃcult: The cor- resp onding decision problem is NP-hard. Although there are some nice theoret- ical results and some work on practical computation of the treewidth of small graphs [Bo d06, BGK08], no eﬃcien t algorithm that computes an optimal tree decom- p osition is kno wn. Notably , it is an op en question whether there is a constant-factor appro ximation for determining the treewidth of a graph [Bo d08]. The second reason is that there is no simple ex plicit analogy to treewidth on directed graphs: A concept of “directed treewidth” was develop by Johnson et al. [JRST01]. Indep enden tly , the concept “D AG-width” w as introduced in [BDHK06] as well as in [Ob d06]. Alternativ ely , another concept “d-width” was in tro duced b y Sa- fari [Saf05]. Finally , a fourth concept “Kelly-width” was proposed b y Hunter et al. [HK07]. Although there is a tendency to “Kelly-width” [MA TV10], it is not even clear whic h of these concepts is the b est analogy to undirected treewidth. The third reason is that there are also problems that are not ﬁxed-parameter tractable with resp ect to the parameter treewidth. Examples are Cap acit a ted Ver tex Cover and Cap a cit a ted Domina ting Set [DLSV08]. A weak er mea- sure of the “degree of acyclicit y” ma y lead to a parameter that allows for ﬁxed- parameter tractability . T o the b est of our kno wledge, there w as no w ork with main fo cus on the parameterized complexity with resp ect to parameters that “measure 22 Goal: Removing the resp ective set makes the graph acyclic Directed graph Undirected graph Remo ving vertices: F eedback v ertex set Remo ving arcs: F eedback arc set Remo ving vertices: F eedback v ertex set Remo ving edges: F eedback edge set Figure 3.1: Ov erview: W e consider four types of feedback sets. The corresp onding parameters are the cardinalities of these sets. the distance of the input graph to an acyclic graph” so far. Informally sp eaking, a fe e db ack set is a substructure of the graph such that its deletion makes the graph acyclic. Dep ending on the type of input (directed or undi- rected graph) and on the structural elements (v ertices or edges/arcs), we consider four parameters (see Figure 3.1). W e start with the formal deﬁnition of the directed case: Let G = ( V , A ) be a directed graph. One calls a subset of vertices V 0 ⊆ V (dir e cte d) fe e db ack vertex set if G − V 0 is acyclic. A subset of arcs A 0 ⊆ A is called fe e db ack ar c set if G 0 := ( V , A \ A 0 ) is acyclic. Analogously , let G = ( V , E ) b e an undirected graph. One calls a subset of v ertices V 0 ⊆ V (undir e cte d) fe e db ack vertex set if G − V 0 is acyclic. A subset of edges E 0 ⊆ A is called fe e db ack e dge set if G 0 := ( V , E \ E 0 ) is acyclic. Determining the sizes of these feedbac k sets leads directly to the following decision problems: Undirected Feedba ck Ver tex Set Given: An undirected graph G = ( V , E ) and an integer k ≥ 1. Question: Is there an undirected feedback vertex set of size at most k ? Undirected Feedback Ver tex Set is NP-complete [Kar72]. With determin- istic ﬁxed-parameter algorithms it can b e solv ed in O (5 k · k · n 2 ) time [CFL + 08]. There is a randomized algorithm which solves Undirected Feedback Ver tex Set in O ( c · 4 k · k n ) time by ﬁnding an undirected feedback vertex set of size k with probabilit y at least 1 − (1 − 4 − k ) c 4 k for an arbitrary constan t c . Directed Feedba ck Ver tex Set Given: A directed graph G = ( V , A ) and an in teger k ≥ 1. Question: Is there a directed feedback vertex set of size at most k ? Directed Feedba ck Ver tex Set is also NP-complete. This follo ws directly from the reduction given by Karp in [Kar72]. How ev er, recent studies prov ed its ﬁxed-parameter tractabilit y , showing that it can b e solved in O (4 k · k ! · n 4 · k 3 ) time [CLL + 08]. Feedba ck Ar c Set Given: A directed graph G = ( V , A ) and an in teger k ≥ 1. Question: Is there a feedback arc set of size at most k ? 3 F eedback sets 23 u 1 u 2 u 3 u 4 u 5 u 6 u 7 d 1 d 2 d 3 d 4 d 5 d 6 d 7 Figure 3.2: F eedbac k sets examples. Here we ha ve an undirected and a directed graph. In the undirected graph there is a feedback v ertex set of size one ( { u 2 } ) and a feedback edge set of size tw o (for example {{ u 2 , u 5 } , { u 6 , u 7 }} ). In the directed graph there is a feedbac k vertex set of size tw o ( { d 1 , d 2 } ) and a feedbac k arc set of size three (for exam- ple { ( d 2 , d 3 ) , ( d 2 , d 5 ) , ( d 1 , d 6 ) } ). The directed Feedback Arc Set is NP-complete [Kar72]. It w as shown b y Ev en et al. [ENSS98] that Feedback Ar c Set and Directed Feedback Ver tex Set can be reduced in linear time from one to eac h other retaining the parameter “size of a feedback set”. Hence, Feedback Arc Set can also b e solved in O (4 k · k ! · n 4 · k 3 ) time. Feedba ck Edge Set Given: A undirected graph G = ( V , E ) and an integer k ≥ 1. Question: Is there a feedback edge set of size at most k ? The undirected Feedback Edge Set is p olynomial-time solv able. It is easy to see that a (minimum) feedback edge set can b e found b y depth-ﬁrst searc h. One just has to ﬁnd a spanning tree. A minimal feedback edge set consists of the edges that are not in this tree. In the follo wing, w e alw a ys talk ab out fe e db ack vertex sets and the problem Feed- ba ck Ver tex Set in b oth cases, the directed and the undirected, whenever it is clear which of them is considered. F urthermore, the parameters are called “size of a feedbac k vertex set” and so on. Of course, in most cases it is useful to know the size of the resp ectiv e minimum feedback set. Ho wev er, the algorithms do not dep end on the minimality of the parameter v alue. This is an adv an tage when we use heuristics or appro ximation algorithms to ﬁnd the feedbac k sets. There are cases where one has to know a concrete feedback set and the running time dep ends on the size of this set (see Section 5.4.1, Section 5.4.2, and Section 6.2). In con trast, there are also cases where the size of the feedback set is used indirectly to prov e the w orst-case running time (see Section 4.3, Section 5.2, and Section 5.3.2). There, it is not even necessary to kno w any feedback vertex set. Some results can b e carried o ver from one parameter to the other: Deﬁnition 3. L et a and b denote two p ar ameters. If a ≤ b for e ach instanc e, then one says a is str onger than b (and b is we aker than a ). 24 Hardness results like W[ t ]-hardness for some p ositiv e in teger t or non-existence of a problem k ernel can b e carried ov er from the weak er to the stronger parameter. In contrast, results like mem b ership to a parameterized complexit y class can b e carried o ver from the stronger to the weak er parameter. It is easy to see that the treewidth of a graph is at most the size of a feedback vertex set of the same graph. F urthermore, for each feedback edge set resp ectiv ely for each feedback arc set there is a feedback vertex set of at most the same size. Hence, “treewidth” is a stronger parameter than “size of a feedbac k vertex set” which is a stronger parameter than “size of a feedbac k edge set” resp ectiv ely “size of a feedback arc set”. 4 Minimum Indegree Deletion In this c hapter, w e analyze the parameterized complexity of Minimum Indegree Deletion . The problem is motiv ated as graph problem corresp onding to construc- tiv e control by deleting candidates for Llull voting (see Chapter 2). The inputs are a directed graph D = ( W, A ), a distinguished v ertex w c ∈ W , and an integer k ≥ 1. The question is whether there is a subset W 0 ⊆ W \ { w c } of size at most k suc h that w c is the only vertex that has minimum indegree in D − W 0 . In most scenarios, computational hardness for control of a v oting rule is a desirable attribute. How ever, NP-hardness should only be a ﬁrst step in this regard, b ecause the hardness does not necessarily hold for sp ecial cases of the input which can b e t ypically in real-world instances. A parameterized analysis will help to discuss computational hardness for sp ecial types of input to the v oting rule. W e present intractabilit y and tractability results for the parameters s v :=“size of a feedbac k vertex set”, s a :=“size of a feed- bac k arc set”, k :=“num b er of vertices to remo ve”, and d :=“maxim um degree of a v ertex” and their combinations. Our results are summarized in Figure 4.1. 4.1 Kno wn results W e brieﬂy discuss previous result [BU09]: There is a simple p olynomial-time al- gorithm that solv es Minimum Indegree Deletion in acyclic directed graphs. In contrast, Minimum Indegree Deletion is W[2]-complete with resp ect to k :=“size of a solution set”. This ev en holds if the input graph is a tournament. P arameterized complexity: single parameter com bined with k s v W[2]-hard, in XP FPT s a W[2]-hard, in XP FPT k W[2]-complete d FPT FPT parameter description s v size of a feedbac k vertex set s a size of a feedbac k arc set k size of a solution set d maxim um degree of a vertex Figure 4.1: Ov erview of the parameterized complexity of Minimum Indegree Deletion . New results are in b oldface. The remaining results are obtained from [BU09]. 26 With resp ect to the parameter d :=“maxim um indegree of a vertex” Minimum In- degree Deletion is ﬁxed-parameter tractable. Inspired by the result for acyclic graphs we study parameters measuring the “degree of acyclicity” of the input graph in the follo wing. 4.2 F eedback vertex/a rc set size as pa rameter Here, we show that Minimum Indegree Deletion is W[2]-hard with respect to the parameter s v :=“size of a feedback v ertex set” and the parameter s a :=“size of a feedbac k arc set”. T o this end, w e need the concept of “domination in an undirected graph”. Deﬁnition 4. L et G = ( V , E ) b e an undir e cte d gr aph. We say that a vertex d dominates a vertex v if d = v or d is a neighb or of v . A subset D ⊆ V is c al le d dominating set if every vertex in V is dominate d by a vertex in D . In the follo wing, w e provide parameterized reductions from Domina ting Set , whic h is W[2]-complete with resp ect to the parameter k :=“size of the dominat- ing set”. Domina ting Set Given: An undirected graph G = ( V , E ) and an integer k ≥ 1. Question: Is there a dominating set of size at most k ? P arameterized reduction. The follo wing reduction is illustrated in Figure 4.2. Giv en a Domina ting Set instance ( G ∗ = ( V ∗ , E ∗ ) , k ) with V ∗ = { v ∗ 1 , v ∗ 2 , . . . , v ∗ n } , w e construct a directed graph G with feedback v ertex set size k + 1 and feedback arc set size ( k + 1) 2 suc h that ( G, w c , n − k ) is a yes-instance of Minimum Indegree Deletion if and only if ( G ∗ , k ) is a y es-instance of Domina ting Set . This means w e hav e a p arameterized reduction if we can do the construction in p olynomial time. The v ertex set of G consists of w c and the union of the following disjoin t vertex sets: • V , D , each con taining one v ertex for every vertex in V ∗ , – V := { v i | i ∈ { 1 , . . . , n }} represen ts the set of “dominating vertices” whic h is illustrated by the no des v 1 , v 2 and v n in Figure 4.2, – D := { d i | i ∈ { 1 , . . . , n }} represents the set of “dominated v ertices” whic h is illustrated by the no des d 1 , d 2 and d n in Figure 4.2. The arcs b etw een v ertices in V and D ensure that “undominated vertices in a solution graph w ould hav e a lo wer indegree than w c ”. They are illustrated by dotted arro ws in Figure 4.2. • X , Y , Z , each con taining k + 1 vertices, – X := { x i | i ∈ { 1 , . . . , k + 1 }} , – Y := { y i | i ∈ { 1 , . . . , k + 1 }} , – Z := { z i | i ∈ { 1 , . . . , k + 1 }} . 4 Minimum Indegree Deletion 27 w c d 1 d 2 . . . d n v 1 v 2 . . . v n One arc from v i to d j ⇔ vertex v ∗ i dominates v ertex v ∗ j in the Dom- ina ting Set instance s 1 , 1 s 1 , 2 . . . s 1 , 6 n s 2 , 1 s 2 , 2 . . . s 2 , 6 n s n, 1 s n, 2 . . . s n, 6 n y 1 y 2 . . . y k +1 x 1 x 2 . . . x k +1 z 1 z 2 . . . z k +1 One arc from eac h v ertex in X to eac h vertex in Y , one from each v ertex in Y to eac h vertex in Z and one from each vertex in Z to eac h vertex in X . B1: outgoing arcs from each v ertex B2: exactly k + 1 ingo- ing arcs to each v ertex B3: exactly k ingoing arcs to eac h vertex B5: exactly k ingoing arcs to eac h vertex B4: at least n outgo- ing arcs from eac h v er- tex Figure 4.2: Minimum Indegree Deletion instance obtained from a parameter- ized reduction from a Domina ting Set instance. 28 start set end set { x 1 , . . . , x k } { s i,j | i ∈ { 1 , . . . , n } and j ∈ { 1 , . . . , n }} { x 2 , . . . , x k +1 } { s i,j | i ∈ { 1 , . . . , n } and j ∈ { n + 1 , . . . , 2 n }} { y 1 , . . . , y k } to { s i,j | i ∈ { 1 , . . . , n } and j ∈ { 2 n + 1 , . . . , 3 n }} { y 2 , . . . , y k +1 } { s i,j | i ∈ { 1 , . . . , n } and j ∈ { 3 n + 1 , . . . , 4 n }} { z 1 , . . . , z k } { s i,j | i ∈ { 1 , . . . , n } and j ∈ { 4 n + 1 , . . . , 5 n }} { z 2 , . . . , z k +1 } { s i,j | i ∈ { 1 , . . . , n } and j ∈ { 5 n + 1 , . . . , 6 n }} Figure 4.3: Concrete assignment of the arcs that are incident to v ertices in S n i =1 S i . F or eac h row of the table there is one arc from eac h vertex from the start set to eac h vertex from the end set. These v ertex sets are designed to “increase the indegree of vertices without increasing the size of a (minimum) feedbac k vertex set”. W e will see that each of them is a feedbac k vertex set and every cycle in G passes trough at least one v ertex in X , Y , and Z . The set X is illustrated by the no des x 1 , x 2 , and x k +1 , Y is illustrated by y 1 , y 2 , and y k +1 , and Z is illustrated by z 1 , z 2 , and z k +1 in Figure 4.2. • S 1 , . . . , S n eac h containing 6 n vertices. These v ertex sets are designed to ensure that the set of remo ved v ertices corre- sp onding to any y es-instance ( G, w c , n − k ) of Minimum Indegree Deletion only contains vertices of V . W e will show that remo ving vertices that are not in V would b e “punished by b eing forced to remov e more than n further ver- tices in S n i =1 S i ”. Eac h set S i is illustrated by the no des s i, 1 , s i, 2 , and s i, 6 n in Figure 4.2. The arcs of G are designed as follo ws: • There is an arc from v i to d j if and only if v ∗ i dominates v ∗ j . These arcs are illustrated b y dotted arrows in Figure 4.2. • There is an arc from each v ertex in V to w c . These arcs are illustrated by thin arro ws from the no des v 1 , v 2 , and v n to the no de w c in Figure 4.2. • There is an arc from each v ertex in X to each v ertex in Y , from each v ertex in Y to each vertex in Z , and from each v ertex in Z to eac h vertex in X . These arcs are illustrated by thin arro ws b etw een the no des x 1 , x 2 , x k +1 , y 1 , y 2 , y k +1 , z 1 , z 2 , and z k +1 in Figure 4.2. • There are k arcs from arbitrary vertices in X ∪ Y ∪ Z to each vertex in D and k + 1 arcs from arbitrary vertices in X ∪ Y ∪ Z to each vertex in V . Outgoing arcs are illustrated b y fat lines from the no des x 1 , x 2 , x k +1 , y 1 , y 2 , y k +1 , z 1 , z 2 , and z k +1 to the b o x B1. Ingoing arcs are illustrated by t w o fat lines from the b ox B1 to the b o xes B2 and B3 and by fat arrows from the b o xes B2 and B3 to the no des v 1 , v 2 , v n , d 1 , d 2 , and d n in Figure 4.2. 4 Minimum Indegree Deletion 29 • F or each i ∈ { 1 , . . . , n } there is an arc from d i and further k outgoing arc from v ertices in X ∪ Y ∪ Z to each vertex in S i suc h that ev ery vertex in X ∪ Y ∪ Z has at least n outneigh b ors in S n i =1 S i . Remo ving a vertex in X ∪ Y ∪ Z decreases the indegree of its n outneighbors in S n i =1 S i to o muc h. A concrete arc assignmen t is giv en in Figure 4.3. The outgoing arcs from vertices in D are illustrated by thin arrows from the no des d 1 , d 2 , and d n to the no des s 1 , 1 , s 1 , 2 , s 1 , 6 n , s 2 , 1 , s 2 , 2 , s 2 , 6 n , s n, 1 , s n, 2 , and s n, 6 n . The outgoing arcs from v ertices in X ∪ Y ∪ Z are illustrated by a fat line from B1 to B4, a fat line from B4 to B5, and fat arro ws from the b ox B5 to the no des s i, 1 , s i, 2 , and s i, 6 n for i ∈ { 1 , . . . , n } in Figure 4.2. The construction ensures that an optimal solution deletes n − k vertices from V and no other v ertex. This will b e pro ved in detail later on, but it is necessary for the further argumen tation to inv estigate the indegrees of the vertices: The indegree of w c is n and the indegree of each v ertex in V is k + 1. Since each v ertex v ∗ i dominates itself, the vertices in D ha ve indegree at least k + 1. Every v ertex in X ∪ Y ∪ Z has trivially also indegree k + 1. What remains are the vertex sets S 1 , . . . , S n . The v ertices in S n i =1 S i ha ve also indegree k + 1. In the following w e give an in tuition wh y they are useful: W e w an t to ensure that if ( G, w c , n − k ) is a y es-instance of Minimum Indegree Deletion , then ev ery solution set only contains vertices of V and esp ecially no v ertices of D . F or eac h v ertex d i with i ∈ { 1 , . . . n } there is a set of v ertices S i := { s i, 1 , . . . s i, 6 n } with an outgoing arc to each of the vertices in S i . This realizes the “punishment”: Removing d i is not p ossible without removing all v ertices in S i (whic h are more than n ). A similar argumentation holds for the v ertices in X ∪ Y ∪ Z , b ecause eac h of them has at least n outneigh b ors in S n i =1 S i . This ﬁnishes the description of the construction. No w, we giv e several lemmata and observ ations to pro ve the correctness of the construction, that is, ( G ∗ , k ) is a y es-instance of Domina ting Set if and only if ( G, w c , n − k ) is a y es-instance of Minimum Indegree Deletion . Lemma 1. If ( G ∗ , k ) is a yes-instanc e of Domina ting Set , then ( G, w c , n − k ) is a yes-instanc e of Minimum Indegree Deletion . Pr o of. Let ( G ∗ , k ) b e a yes-instance and V ∗ d ⊆ V ∗ b e a dominating set of size k for G ∗ . No w, we delete a vertex v i ∈ V from G , if v ∗ i / ∈ V ∗ d . Since V ∗ d is of size k and V of size n , we deleted n − k v ertices. The v ertex w c has no w indegree k . W e hav e to show that eac h other vertex has indegree at least k + 1. By construction every v ertex in G has indegree at least k + 1. Since we remo v ed only v ertices in V , the only vertices which can hav e a decreased indegree are v ertices in D . (These are the only vertices which can b e outneigh b ors of a vertex in V .) Due to the fact, that V ∗ d is a dominating set, every vertex in D keeps at least one inneigh b or in V . By construction there are k further inneigh b ors in X ∪ Y ∪ Z . Hence, eac h vertex in D has at least indegree k + 1. T rivially , all other v ertices keep indegree at least k + 1, b ecause we did not remov e any inneigh b or. Th us, w c has minimum indegree and ( G, w c , n − k ) is a yes-instance of Minimum Indegree Deletion . Lemma 1 show ed the direction from left to righ t of the parameterized equiv alence b et ween the Domina ting Set solution and the Minimum Indegree Deletion 30 solution; now, we show the rev erse direction. Consider ( G, w c , n − k ) resulting from the parameterized reduction. Let M d b e a solution set for ( G, w c , n − k ). Several observ ations are very useful for the further argumentation. Observ ation 1. The c onstructe d gr aph G has a fe e db ack vertex set with at most k + 1 vertic es and a fe e db ack ar c set with at most ( k + 1) 2 ar cs. Pr o of. W e show b y con tradiction that G − X is acyclic. Assume that there is a (non-empt y) cycle C = ( c 1 , . . . , c l ). Since w c and each v ertex in S n i =1 S i has no outneigh b or, neither w c nor an y vertex in S n i =1 S i can b e part of C . Eac h vertex in d i ∈ D has only outneigh b ors in S i . Th us, C con tains no v ertex from D . V ertices from V hav e only outneigh b ors in { w c } ∪ D which implies that C do es not contain an y vertex from V . V ertices from Z hav e only outneighbors in S n i =1 S i ∪ V ∪ D , a set that con tains no v ertex in C . The remaining v ertices from Y hav e only outneigh b ors in Z ∪ V ∪ D S n i =1 S i . Th us, C must b e empty; a contradiction. Observ ation 2. It holds that w c has inde gr e e at le ast k in G − M d . Pr o of. Assume that w c has indegree at most k − 1 in G − M d . Since w c has indegree n in the original graph G , we must ha v e deleted n − k + 1 inneighbors of w c . Hence, M d is a solution set of size at least n − k + 1; a conﬂict. Observ ation 3. The solution set M d do es not c ontain vertic es of D , X , Y , or Z . Pr o of. Assume that u ∈ M d is a v ertex from D ∪ X ∪ Y ∪ Z . By construction of G , vertex u has at least n outneigh b ors in S n i =1 S i . W e call them S -outneighbors in the following. After removing u , every S -outneigh b or must hav e indegree at most k in G − M d , b ecause it has indegree exactly k + 1 in G (by construction). Since the ﬁnal degree of w c is at least k (see Observ ation 2) and w c is the only v ertex with minim um indegree in G − M d , every S -outneighbor m ust b e also in M d . Thus, M d has size at least n + 1; a conﬂict. Observ ation 4. In G − M d , the distinguishe d vertex w c has inde gr e e exactly k . Pr o of. Assume that w c has an indegree at least k + 1. Since w c has to b e the only v ertex with minim um indegree, we hav e to delete each vertex with an indegree of at most k + 1. Remember that each vertex in X ∪ Y ∪ Z has an indegree of exactly k + 1. Ho wev er, no v ertex in X ∪ Y ∪ Z is part of the solution set (see Observ ation 3); a conﬂict. Observ ation 5. It holds that M d only c ontains vertic es fr om V . Pr o of. In the original graph G the distinguished v ertex w c has indegree n . Due to Observ ation 4 the solution set M d m ust contain at least n − k inneigh b ors of w c . Due to the assumption that M d has size at most n − k there is no other v ertex in M d . Lemma 2. If ( G, w c , n − k ) is a yes-instanc e of Minimum Indegree Deletion , then ( G ∗ , k ) is a yes-instanc e of Domina ting Set . 4 Minimum Indegree Deletion 31 Pr o of. W e show that if M d is a solution set, then there is a dominating set V ∗ d ⊆ V ∗ with at most k vertices. One can construct a dominating set V ∗ d with | V ∗ d | = k as follo ws: F or each vertex v i ∈ V \ M d add vertex v ∗ i to V ∗ d . Due to Observ ations 4 and 5, V ∗ d has size k . It remains to show that V ∗ d is a domination set. Assume that there is a vertex v ∗ f that is not dominated by any v ertex in V ∗ d . Thus, neither v f nor an y v d ∈ N ( v f ) is in M d . Due to Observ ation 3, no v ertex in D was deleted. Due to the construction of G there is no arc from an y v ertex in V to the undominated v ertex d f . Hence, d f has indegree of k and w c is not the only vertex with minimum indegree; a con tradiction. Putting all together, w e arrive at the following theorem: Theorem 1. Minimum Indegree Deletion is W[2]-har d with r esp e ct to the p ar ameter s v as wel l as with r esp e ct to the p ar ameter s a . Pr o of. W e show that the transformation from ( G ∗ , k ) to ( G, w c , n − k ) is a pa- rameterized reduction: By construction, the transformation can be executed in f ( k ) · p oly ( | x | ) time with f ( k ) b eing a function only dep ending on k and | x | b eing the size of the input. Due to Observ ation 1 the parameter s v ( s a ) is b ounded by a function only dep ending on k . The equiv alence follows Lemma 1 and Lemma 2. Theorem 1 pro vides a relativ e low er b ound for the parameterized complexity with resp ect to the feedbac k set parameters s v and s a . An upp er b ound, namely the mem b ership in XP, is a corollary of the main result of the next section. 4.3 F eedback vertex set size and solution size as combined pa rameter T aking s v :=“size of a feedback vertex set” ( s a :=“size of a feedback arc set”) as pa- rameter Minimum Indegree Deletion do es not yield ﬁxed-parameter-tractability . As mentioned in Section 4.1, k :=“num b er of vertices to delete” as a parameter also leads to W[2]-hardness. So, it makes sense to consider the combined parameters ( s v , k ) as well as ( s a , k ) for Minimum Indegree Deletion . In this section, we sho w that each of these com bined parameters leads to a ﬁxed-parameter algorithm. The algorithm describ ed in the following gets as input a directed graph, a distin- guished v ertex and a p ositive integer k and outputs a solution set of size k . Hence, with an additional factor k to the running time it also solves the corresp onding minimization problem, where one asks for a minimum-size solution. The algorithm do es not need to kno w or compute s v or ev en a feedback vertex set. W e start with lo oking at the acyclic sp ecial case to describ e the basic idea of the algorithm. One motiv ation to inv estigate Minimum Indegree Deletion with resp ect to the parameters that measure the “degree of acyclicit y” is that the problem is NP-hard in general but p olynomial-time solv able in the acyclic sp ecial case [BU09]. The corresp onding algorithm is based on the fact that a directed acyclic graph alwa ys con tains a vertex with indegree zero. It is an exhaustiv e application of the follo wing step: If there is a v ertex ( 6 = w c ) with indegree zero, then remov e it. This is trivially correct, since w e wan t that w c is the only v ertex with minimum indegree (zero). 32 w c b 1 b 2 b 3 b 4 b 5 a 1 a 2 a 3 a 4 Figure 4.4: Directed cyclic graph where a minimum-size solution set of Minimum Indegree Deletion consists of b 1 and b 3 whereas a 2 , a 3 , and a 4 ha ve minim um indegree of 1. There are tw o reasons wh y we cannot apply this algorithm to general directed graphs: 1. Adapting the idea directly b y “remo ving ev ery indegree-zero vertex except w c ” is not correct, b ecause it is p ossible that there is no v ertex with indegree zero. 2. Adapting the idea in a more generalized wa y as “removing ev ery vertex with minim um indegree except w c ” is not correct. Unfortunately , it is even p ossible that we do not need to remo ve all or ev en any vertex that has minimum indegree in the input graph. The ﬁrst p oin t is trivial. The second p oint is illustrated in Figure 4.4. As w e see in this example another strategy to make w c the only vertex with minim um indegree (b esides removing vertices with smaller indegree) is to remo ve inneighbors of w c to di- rectly decrease its indegree. This leads to the algorithm MinimumIndegreeDeletion in Figure 4.5: T o analyze the algorithm we take a closer lo ok at our directed graph instance. First, we see that the minim um indegree in the solution graph is b ounded b y the size of any feedback vertex set. Lemma 3. L et G = ( V , E ) denote a dir e cte d gr aph. L et d min denote the minimum inde gr e e of the vertic es in V and let V f ⊆ V b e a fe e db ack vertex set. Then, | V f | ≥ d min . Pr o of. There exists a vertex v z with indegree zero in G − V f , b ecause G − V f is acyclic. Since G has no v ertex with indegree smaller than d min , the feedback vertex set V f m ust contain at least d min inneigh b ors of v z . Th us, | V f | ≥ d min . A consequence is that the ﬁnal indegree of w c in a solution graph is at most the size of a feedback v ertex set s v . This holds due to Lemma 3 and the fact that a feedback v ertex set in G implies a feedback v ertex set in G − M d of at most the same size. Hence, the lo op in line 2 of the algorithm MinimumIndegreeDeletion is deﬁned correctly . The main part (lines 4-14) of MinimumIndegreeDeletion is exhaustive exploration of the searc h space. It remains to show that the condition in line 3 is correct. Assume to w ards a contradiction that | N ( w c ) | ≥ k + i in a yes-instance of Minimum Indegree Deletion . Remo ving all but i neigh b ors implies a solution 4 Minimum Indegree Deletion 33 1: pro cedure MinimumIndegreeDeletion 2: for each i := 0 to s v do 3: if | N ( w c ) | ≤ i + k then 4: for each size- i subset U ⊆ N ( w c ) do 5: Remo ve D := N ( w c ) \ U from G . 6: M d := D 7: while there is a vertex d 6 = w c with indegree at most i do 8: Remo ve d from G . 9: M d := M d ∪ { d } 10: end while 11: if | M d | ≤ k then 12: return M d 13: end if 14: end for 15: end if 16: end for 17: return “no”; 18: end pro cedure Figure 4.5: Fixed-parameter algorithm that solves Minimum Indegree Deletion with resp ect to the parameter ( s v , k ). The v ariable i represents the ﬁnal indegree of w c in the solution graph. set of size greater than k ; a contradiction. Hence, the algorithm is correct and w e arriv e at the following theorem: Theorem 2. Minimum Indegree Deletion with r esp e ct to the c ombine d p ar am- eter ( s v , k ) , with s v b eing the size of a fe e db ack vertex set and k b eing the size of a solution set, is solvable in O ( s v · ( k + 1) s v · n 2 ) time. Pr o of. The correctness of the algorithm was already sho wn. It remains to analyze the running time. In the worst case, w e hav e to start at most s v times with the ﬁrst (out-most) lo op. In the second lo op, w e try at most  s v + k k  subsets. Th us, we hav e  s v + k k  = ( s v + k )! k ! · ( s v + k − k )! = Q s v i =1 k + i s v ! ≤  k + 1 1  s v subsets. The third lo op can b e done in O ( n 2 ) time. Lo oking at the second lo op of the algorithm helps us to see that Minimum Inde- gree Deletion with resp ect to the single parameter s v is actually at least in XP. Branc hing into all p ossible subsets of i ≤ s v inneigh b ors of w c is of course p ossible in O ( n i ) time. In a more formal w ay , we arrive at the following corollary: Corollary 1. The pr oblem Minimum Indegree Deletion with r esp e ct to the p ar ameter s v := “size of a fe e db ack vertex set” is in XP . Pr o of. Since ( k + 1), n , and s v are upp er b ounded by the input size, this follows di- rectly from Theorem 2. The running time of MinimumIndegreeDeletion is b ounded b y O ( | x | s v +3 ) with | x | b eing the input size. 5 Minimum Degree Deletion In this section, w e inv estigate the parameterized complexit y of Minimum Degree Deletion which can b e considered as undirected v arian t of Minimum Indegree Deletion . It mo dels electoral control by remo ving candidates of a sp ecial v oting rule, see Section 2.2. In Chapter 4, we show ed ﬁxed-parameter intractabilit y of Minimum Indegree Deletion with resp ect to the parameters s v :=“size of a feedbac k vertex set” and s a :=“size of a feedback arc set”, resp ectively . In con trast, w e show in this c hapter ﬁxed-parameter tractability ev en for the stronger parameter “treewidth of the input graph”. F urthermore, w e diﬀerentiate the parameters by comparing their k ernel sizes. Firstly , w e show that Minimum Degree Deletion is as well as its directed v arian t Minimum Indegree Deletion ﬁxed-parameter in tractable with resp ect to the parameter k :=“num b er of v ertices to delete”. An o verview ab out the kernel sizes and the parameterized complexit y with resp ect to the (com bined) parameters is given in Figure 5.1. P arameterized complexity: parameter description complexit y t w treewidth of the input graph FPT s v size of a feedbac k vertex set FPT s a size of a feedbac k arc set FPT k size of a solution set W[1]-hard, in XP d maximum degree of a vertex FPT Kernel sizes: single parameter com bined with k t w no p olynomial no p olynomial s v no p olynomial no p olynomial s a v ertex-linear v ertex-linear k no k ernel d op en open Figure 5.1: Ov erview of the parameterized complexit y of Minimum Degree Dele- tion and the corresp onding kernel sizes. New results are in b oldface. The results for the parameter d :=“maximum degree of a v ertex” can b e directly transferred from the results for the directed v arian t in [BU09]. A v ertex-linear kernel is a problem k ernel whose size is linear in the n umber of vertices. 36 5.1 Solution size as pa rameter In this section, we analyze the parameterized complexit y of Minimum Degree Deletion with resp ect to the parameter k :=“n umber of vertices to delete”. Using metho ds similar to those in Section 4.2 we show W[1]-hardness b y presen ting a parameterized reduction from Independent Set with resp ect to the parameter “indep enden t set size”. Below, w e give a description and illustration of the reduction. W e ﬁrst sho w that one can assume that each Independent Set instance has an ev en num b er of edges. Let G = ( V , E ) b e an undirected graph with n := | V | and m := | E | . One can transform eac h instance with an o dd v alue of m to a new instance with an even v alue of m suc h that the new instance is a y es-instance if and only if the original instance is a yes-instance and the parameter v alue do es not c hange. T o this end, w e hav e to consider tw o cases: The ﬁrst case is that the n umber of vertices is o dd. Then w e only hav e to add a new vertex and connect it to each v ertex of the original graph. The second case is that the num b er of vertices is ev en. Here, w e hav e to add a clique of three new vertices and connect each of these vertices of eac h v ertex of the original graph. So, we get 3 n + 3 new edges whic h is o dd since n is ev en. T rivially , none of the new v ertices will ev er b e a part of an indep endent set of size at least t wo 1 in b oth cases. P arameterized reduction. The follo wing reduction is illustrated in Figure 5.2. Giv en an Independent Set instance ( G ∗ = ( V ∗ , E ∗ ) , k ), with V ∗ = { v ∗ 1 , v ∗ 2 , . . . , v ∗ n } and E ∗ = { e ∗ 1 , e ∗ 2 , . . . , e ∗ m } , w e construct an undirected graph G , with a distinguished v ertex w c suc h that ( G, w c , k ) is a yes-instance of Minimum Degree Deletion if and only if ( G ∗ , k ) is a yes-instance of Independent Set . The v ertex set of G consists of w c and the union of the follo wing disjoint vertex sets: • V containing one v ertex for ev ery v ertex in V ∗ and E containing one v ertex for ev ery edge in E ∗ . – V := { v i | i ∈ { 1 , . . . , n }} represen ts the set of v ertices which is illustrated b y the no des v 1 , v 2 and v n in Figure 5.2. – E := { e i | i ∈ { 1 , . . . , m }} represents the set of “connections b etw een v ertices” which is illustrated by the no des e 1 , e 2 , and e m in Figure 5.2. • F or eac h vertex v i ∈ V there are t wo cliques C v i and C i of size n − k + 1, and for each v ertex e i ∈ E there is one clique C e i of size n − k . These cliques are illustrated b y stars in Figure 5.2. The edges (b esides inner-clique edges) of G are dra wn as follo ws: • There is an edge b etw een v i and e j if and only if v ∗ i is incident to e ∗ j . These edges are illustrated b y dotted lines in Figure 5.2. • The distinguished v ertex w c is connected to each vertex in V . The corresp ond- ing edges are illustrated by thin lines b et w een the no de w c and the no des v 1 , v 2 , and v n in Figure 5.2. 1 Ev ery graph with at least one v ertex has an indep endent set of size one. 5 Minimum Degree Deletion 37 w c v 1 v 2 . . . v n C v 1 C v 2 . . . C v n C 1 C 2 . . . C n e 1 e 2 . . . e m C e 1 C e 2 . . . C e m Exactly n − k v ertices in C x with x ∈ { v 1 , . . . , v n } ∪ { e 1 , . . . , e m } are connected to the v ertex x . Eac h vertex from C e j with j ∈ { 1 , 3 , 5 , . . . , m − 1 } is connected to exactly one vertex from C e j +1 . There is an edge b etw een e x and v y if and only if v ∗ y is inciden t to e ∗ x Eac h vertex from C v i with i ∈ { 1 , 2 , . . . , n } is con- nected to exactly one vertex from C i . n − k n − k n − k n − k + 1 n − k + 1 n − k + 1 n − k n − k n − k n − k Figure 5.2: Minimum Degree Deletion instance obtained from a parameterized reduction from an Independent Set with an ev en num b er of edges. Eac h star C x with x ∈ { v 1 , . . . , v n } represents a clique with n − k v ertices. Eac h star C x with x ∈ { 1 , . . . , n } ∪ { e 1 , . . . , e m } represen ts a clique with n − k + 1 vertices. 38 • Each vertex of ev ery clique C e j with j ∈ { 1 , 3 , . . . , m − 1 } is connected to exactly one (diﬀerent) vertex in C e j +1 . (Note that m − 1 is o dd.) The cor- resp onding edges are illustrated by a fat line betw een C e 1 and C e 2 whic h is lab eled with n − k in Figure 5.2. • Exactly n − k vertices of each clique C x with x ∈ { v 1 , . . . , v n , e 1 , . . . , e m } are connected to the vertex x . The corresp onding edges are illustrated by the remaining fat lines whic h are lab eled with n − k in Figure 5.2. • Each v ertex of every clique C v i with i ∈ { 1 , . . . , n } is connected to exactly one (diﬀeren t) vertex in C i . The corresp onding edges are illustrated by fat lines whic h are lab eled with n − k + 1 in Figure 5.2. Our goal is to ensure that an optimal solution deletes k vertices from V and no other v ertex. Therefore, w e set the degree of w c to n and for eac h other vertex to at least n − k + 1. Each vertex in C i with i ∈ { 1 , . . . , n } and C e j with j ∈ { 1 , . . . , m } has degree n − k + 1 and each vertex in C v i with i ∈ { 1 , . . . , n } has degree n − k + 1 or n − k + 2. Eac h v ertex e j with j ∈ { 1 , . . . , m } has degree n − k + 2 and each vertex v i with i ∈ { 1 , . . . , n } has degree at least n − k + 1. This ﬁnishes the description of the construction. Now, w e give sev eral lemmata and observ ations to pro ve the correctness of the construction, that is, ( G ∗ , k ) is a y es-instance of Independent Set if and only if ( G, w c , k ) is a yes-instance of Minimum Degree Deletion . Lemma 4. If ( G ∗ , k ) is a yes-instanc e of Independent Set , then ( G, w c , k ) is a yes-instanc e of Minimum Degree Deletion . Pr o of. Let ( G ∗ , k ) b e a y es-instance and V ∗ d ⊆ V ∗ a size- k set of indep endent v ertices. W e delete each vertex v i ∈ V from G , if v ∗ i ∈ V ∗ d . Since | V ∗ d | = k and | V | = n , w e deleted k v ertices and the vertex w c has now degree of n − k . It holds that each v ertex e j with j ∈ { 1 , . . . , m } has degree at least n − k + 1, b ecause V ∗ d is an indep enden t set whic h means that at most one of the neighbors from e j in V is deleted. T rivially , the degree of all other vertices is at least n − k + 1. Thus, w c has minim um degree and ( G, w c , k ) ∈ Minimum Degree Deletion . Next, we will show the opp osite direction of the equiv alence. T o this end, several observ ations are very useful for further argumen tation. In the follo wing, let G ∗ = ( V ∗ , E ∗ ) b e an undirected graph and k a p ositive integer. W e apply our reduction and denote the resulting graph and its v ertices and vertex sets as describ ed ab o ve. Let M d denote a set of k v ertices of G such that its deletion makes w c b ecome the only v ertex with minimum degree. Observ ation 6. It holds that w c has de gr e e exactly n − k after deleting M d fr om G . Pr o of. Assume tow ards a con tradiction that the degree of w c do es not equal n − k . First consider the case that w c has degree less than n − k after deleting M d from G . In the original graph w c has degree n . T o reac h degree less than n − k w e ha ve to delete more than k v ertices; a contradiction. Second consider the case that w c has degree more than n − k after deleting M d from G . This means that the degree of w c is at least n − k + 1. Since M d is a solution set, no other vertex with degree n − k + 1 5 Minimum Degree Deletion 39 can exist after deleting M d from G . So, we hav e to delete more than n > k v ertices, since there are more than n cliques with vertices with degree exactly n − k + 1; a con tradiction. Altogether, w c has degree exactly n − k after deleting M d from G . Observ ation 7. It holds that M d only c ontains vertic es of v i with i ∈ { 1 , . . . , n } . Pr o of. The argumen tation is v ery simple. Due to Observ ation 6 it is clear that M d con tains at least k v ertices v i with i ∈ { 1 , . . . , n } . Since M d has size k , there is no other v ertex in M d . Lemma 5. If ( G, w c , k ) is a yes-instanc e of Minimum Degree Deletion , then ( G ∗ , k ) is a yes-instanc e of Independent Set . Pr o of. Let ( G, w c , k ) b e a yes-instance of Minimum Degree Deletion . W e can build a size- k indep endent set V ∗ d as follows: F or each vertex v i ∈ M d add vertex v ∗ i to V ∗ d . Due to Observ ations 6 and 7, V ∗ d has size k . It remains to b e sho wn that V ∗ d is an indep enden t set. Assume that there is an edge e ∗ j with j ∈ { 1 , . . . , m } that is inciden t to an y t wo v ertices v a , v b ∈ V ∗ d . Thus, b oth v a and v b are in M d . Due to the construction of G the v ertex e j has degree of n − k after deleting v a and v b and m ust b e deleted, to o; a con tradiction to Observ ation 7. Altogether, w e arrive at the following: Theorem 3. Minimum Degree Deletion is W[1]-har d with r esp e ct to the p a- r ameter k := “numb er of vertic es to delete”. Pr o of. W e sho w that the transformation from ( G ∗ , k ) to ( G, w c , k ) is a parameterized reduction: By construction, it can b e executed in f ( k ) · p oly( | x, k | ) time. The new parameter equals the original one. The equiv alence follows Lemma 4 and Lemma 5. Theorem 3 pro vides a relativ e low er b ound for the parameterized complexity with resp ect to the parameter k . An upp er b ound, namely the mem b ership in XP, is quite easy to see. Simply c hecking for eac h V 0 ⊆ V whether w c is the only v ertex with minim um degree in G − V 0 already leads to the follo wing: Prop osition 1. Minimum Degree Deletion is in XP with r esp e ct to the p ar am- eter k := “numb er of vertic es to delete”. 5.2 Size of a feedback edge set as pa rameter In Section 4.3, w e show ed that Minimum Indegree Deletion is ﬁxed-parameter- tractable with resp ect to the com bined parameter “size of a feedbac k v ertex/arc set” and “size of a solution set”. In con trast, it is W[2]-hard for b oth single param- eters (Section 4.2 and [BU09]). It is easy to adapt the algorithm of Section 4.3 to the undirected version Minimum Degree Deletion . Hence, w e can show ﬁxed- parameter tractabilit y for the combined parameter “size of a feedback vertex/edge set” and “size of a solution set”. How ev er, the parameterized intractabilit y for the 40 single parameters “size of a feedback vertex set” and “size of a feedbac k arc/edge set” cannot b e transferred as easily . In contrast to the hardness results of the directed problem w e will sho w ﬁxed- parameter tractabilit y for b oth parameters. W e will start with a data reduction rule that ac hieves a vertex-line ar kernel , that is, a problem kernel whose size is linear in the num b er of vertices, for Minimum Indegree Deletion with resp ect to the parameter “size of a feedbac k edge set”. A vertex-linear k ernel. Our k ernelization is based on a simple data reduction rule. The follo wing lemma ensures p olynomial running time for this rule. Lemma 6. L et G = ( V , E ) b e an undir e cte d gr aph and k a p ositive inte ger. In O ( n 2 · k ) time and O ( n 2 + n ) sp ac e one c an determine whether ther e is a set of vertic es M d ⊆ V with | M d | ≤ k such that w c is the only vertex with minimum de gr e e in G − M d and deg( w c ) ≤ 1 . Pr o of. Determining whether there is a solution set M d ⊆ V with | M d | ≤ k such that w c is the only v ertex with minimum degree and deg ( w c ) ≤ 1 works as follows: T o manage the degree information of the vertices we use an adjacency matrix and store the sums of eac h columns and each ro w. The column and row sums give us directly the degree of each vertex. The initialization of the matrix costs O ( n 2 ) time and O ( n 2 + n ) space. In a ﬁrst step w e remo ve all neigh b ors of w c if deg ( w c ) = 0 and all but one neighbor if deg ( w c ) = 1. Subsequently , w e remov e all v ertices with degree at most deg( w c ). Remo ving one vertex costs O ( n ) time, b ecause w e need to up date the matrix. There are at most k remo v al steps. If deg( w c ) = 1 there are up to n p ossible neighbor which are not remo ved. This means an additional factor of O ( n ) in the case deg( w c ) = 1. Thus, w e need O ( n · ( k · n )) = O ( n 2 · k ) time and O ( n 2 + n ) space. Reduction Rule “Remo v e Low Degree” Let G = ( V , E ) b e an undirected graph and k b e a p ositiv e in teger. W e denote b y RLD( G ) the graph resulting by the follo wing data reduction: If there is set of v ertices M d ⊆ V with | M d | ≤ k suc h that w c is the only v ertex with minimum degree and deg ( w c ) ≤ 1 in G − M d , then replace G with a new graph which only contains the single v ertex w c and set the parameter to zero. Otherwise, w c has degree at least t wo in ev ery optimal solution, iterativ ely remo ve eac h vertex with degree at most t wo and decrease the parameter by one in each remo v al step. Due to Lemma 6, Reduction Rule “Remov e Low Degree” can b e executed in p oly- nomial time. It is easy to verify that the rule is sound. The next observ ation follows directly from the construction of RLD( G ). Observ ation 8. Every vertex ( 6 = w c ) in RLD( G ) has de gr e e at le ast thr e e. No w, we are ready to b ound the num b er of vertices in RLD( G ) with the help of this observ ation. F or the ease of argumen tation we ﬁrstly deﬁne a transformation whic h, given a forest, remov es every inner vertex with degree tw o by connecting its neigh b ors. 5 Minimum Degree Deletion 41 Deﬁnition 5. L et G denote a for est. The function DisLined( G ) denotes the r esult of the exhaustive applic ation of the fol lowing pr o c e dur e: If ther e is an induc e d P 3 (p ath of length thr e e) such that the midd le vertex v 0 in P 3 has de gr e e two in G , then r emove v 0 and c onne ct b oth its neighb ors. Theorem 4. Ther e is a 2 s e -vertex kernel for Minimum Degree Deletion with r esp e ct to the p ar ameter s e := “size of a fe e db ack e dge set”. It is c omputable in O ( n 2 · k ) time with n b eing the numb er of vertic es and k b eing the numb er of vertic es to delete. Pr o of. Let G denote an undirected graph. W e show that there are at most 2 · s e v ertices with s e b eing the size of a feedback edge set in RLD( G ). Let E d denote a feedbac k edge set of size s e . Clearly , G − E d is a forest. Since eac h vertex in G has degree at least three (Observ ation 8), each leaf in G − E d m ust b e inciden t to at least t wo edges in E d . It holds that G − E d con tains l ≤ s e lea ves, b ecause each leaf must b e incident to t wo edges of the feedback edge set and each edge of the feedback edge set can b e inciden t to at most t wo leav es. F urthermore, the sum of incidences of the edges in E d is 2 s e . Eac h inner vertex of degree tw o in G − E d m ust b e inciden t to at least one edge in E d . Since there are l leav es in G − E d , only 2 s e − 2 l incidences are left ov er. Hence, G − E d con tains at most 2 s e − 2 l inner vertices with degree t wo. It holds that G − E d has at most l inner vertices with degree at least three if DisLined( G − E d ) has at most l inner vertices with degree at least three. Ev en if DisLined( G − E d ) is a complete binary tree there are at most l / 2 + l / 4 + · · · + 1 = l inner v ertices. Altogether, G has at most l + 2 s e − 2 l + l = 2 s e v ertices. The running time follo ws Lemma 6. A search tree algo rithm. Of course, the 2 s e -v ertex-kernel already pro vides ﬁxed- parameter tractabilit y . The running time of a corresp onding brute-force algorithm is O (4 s e · n 2 ) with s e b eing the size of a feedback edge set. The polynomial factor is for c hecking the correctness of the “guessed” solution set. A simple reﬁnement helps to give the search tree algorithm MDD-search (see Figure 5.3). The input is an instance of Minimum Degree Deletion ( G = ( V , E ) , w c , k ) and a feedback edge set E f with | E f | = s e . W e start with showing the correctness of MDD-search . After applying the data reduction rule (see line 2), MDD-search branc hes o ver each N 0 ⊆ N E with N E b eing the set of w c -neigh b ors that are connected to w c b y an edge of E d (see line 3). Here, the set N 0 expresses the “v ertices from N E that are part of the solution set”. MDD-search remo ves N 0 from the graph, decreases the parameter v alue, and applies the data reduction rule again (see line 4-6). Then, MDD-search branches o ver N 00 ⊆ N ( w c ) \ N E whic h are the “remaining neighbors of w c that are part of the solution set” (see line 7). The algorithm remov es N 00 from the graph and decreases the parameter v alue (see lines 8-9). Finally , MDD-search determines the remaining part of the solution set by iterativ ely removing all v ertices with degree ≤ deg( w c ) (see lines 10-13). Clearly , the algorithm is correct, b ecause it ﬁnally branc hes ov er all neighbors of w c that can b e part of a solution set (see lines 2 and 9) and detects the remaining solution set v ertices which are uniquely determined in eac h branching (see lines 10-13). It remains to analyze the running time. The ﬁrst lo op (line 3) iterates O (2 | N E | ) times. By deﬁnition, | N E | ≤ s e . No w consider the graph G 0 − N E in line 7. Clearly , 42 1: pro cedure MDD-search ( G = ( V , E ), w c , k , E f ) 2: ( G 0 , k 0 ) := RLD( G ) 3: for each N 0 ⊆ N E with N E := { x | { x, w c } ∈ E f } do 4: Remo v e N 0 from G 0 . 5: k 0 := k − | N 0 | 6: ( G 0 , k 0 ) := RLD( G 0 ) 7: for eac h N 00 ⊆ ( N ( w c ) \ N E ) in G 0 do 8: Remo ve N 00 from G 0 . 9: k 0 := k 0 − | N 00 | 10: while there is a vertex v with deg ( v ) ≤ deg ( w c ) in G 0 do 11: Remo ve v from G 0 . 12: k 0 := k 0 − 1 13: end while 14: if k 0 ≥ 0 then 15: return “y es” 16: end if 17: end for 18: end for 19: return “no” 20: end pro cedure Figure 5.3: Fixed-parameter algorithm that solves Minimum Degree Deletion with resp ect to the parameter in O (2 s e · n 2 ) time. G 0 − N E has a feedback edge set E 0 f with | E 0 f | = s e − | N E | . Due to the pro of of Theorem 4 we already kno w that ( G 0 − N E ) − E 0 f has at most | E 0 f | leav es. Since the “remaining neighborho o d of w c in G 0 ”, namely N ( w c ) \ N E in G 0 , do es not change after remo ving N E and E 0 f from G 0 , | N ( w c ) \ N E | is also b ounded b y | E 0 f | . Hence, the second lo op iterates at m ost O (2 | E 0 f | ) = O (2 s e −| N E | ) times. The remaining op erations can b e done in O ( n 3 ) time. Putting all together w e arrive at the following: Theorem 5. Minimum Degree Deletion is solvable in O (2 s e · n 3 ) time with s e b eing the size of a fe e db ack e dge set and n b eing the numb er of vertic es. As already men tioned we also show ﬁxed-parameter tractabilit y for the parameter s v . Since t w :=“treewidth of the input graph” is a lo w er b ound for s v , ﬁxed-parameter tractabilit y for t w w ould trivially imply ﬁxed-parameter tractabilit y for s v . Hence, w e start with the inv estigation of the parameterized complexit y of Minimum De- gree Deletion with resp ect to the parameter t w in the follo wing section. 5.3 T reewidth as pa rameter In the previous sections, we hav e presen ted explicit ﬁxed-parameter algorithms to pro ve that the problems are tractable with resp ect to the corresp onding parame- ters. It is sometimes hard to ﬁnd explicit algorithms that solve a parameterized problem. F ortunately , there are results that state that large classes of problems 5 Minimum Degree Deletion 43 can b e solved in linear time when a tree decomp osition with constan t treewidth is kno wn (see [Cou90, Cou09]). Note that the metho d stated in this section is of purely theoretical in terest. The corresponding running times hav e huge constant factors and com binatorial explosions with resp ect to the parameter treewidth. Hence, for practical applications one should search for an eﬀective problem-sp eciﬁc algorithm. 5.3.1 Monadic second-o rder logic W e use a to ol called monadic se c ond-or der lo gic (or short MSO ). This is an ex- tension to the well-kno wn ﬁrst-order logic by quan tiﬁcation o v er sets. Courcelle’s Theorem [Cou90] says that the veriﬁcation of a graph prop erty is ﬁxed-parameter tractable with resp ect to the parameter treewidth if the prop erty can b e expressed with monadic second-order logic. An extensive ov erview ab out the ﬁeld of monadic second-order logic can b e found in [Cou09]. In the follo wing, we describ e the lan- guage and syn tax of MSO-formulae. An MSO-formula consists of: • an inﬁnite supply of individual variables , b y con ven tion denoted b y small let- ters x , y , z , . . . , • an inﬁnite supply of set variables , by conv en tion denoted b y capital letters X , Y , Z , . . . , • to express graph prop erties t wo unary relations, by conv ention denoted as V and E , and a binary relation, by conv en tion denoted as I , where – the relation V can b e interpreted as “b eing a vertex”, – the relation E can b e interpreted as “b eing an edge”, – and the relation I can b e interpreted as “b eing incident”, • some logical op erators ( ¬ , ∧ , ∨ , → , and ↔ ) and the quantiﬁers ∃ and ∀ . The relations will b e used in preﬁx notation. F or a graph G = ( V , E ) let U := V ∪ E . An assignment α for an MSO-form ula maps eac h individual v ariable to an element of U and eac h set v ariable to a subset of U . One deﬁnes the concept of an assignmen t α satisfying an MSO-formula φ , written ( G, α ) | = φ for a giv en graph G . Now, we can deﬁne the atomic MSO-formulae and their semantics. Let G = ( V , E ) b e a graph, x and y b eing individual v ariables, and X b e a set v ariable. W e hav e the follo wing atomic MSO-formulae: atomic form ula semantics x = y ( G, α ) | = x = y ⇔ α ( x ) = α ( y ) V x ( G, α ) | = V x ⇔ α ( x ) ∈ V E x ( G, α ) | = E x ⇔ α ( x ) ∈ E I xy ( G, α ) | = I xy ⇔ α ( x ) ∈ V is incident to α ( y ) ∈ E X x ( G, α ) | = X x ⇔ α ( x ) ∈ X Moreo ver, all other (more complex) MSO-form ulae can b e inductiv ely built as fol- lo ws: • If φ is an MSO-formula, then ¬ φ is an MSO-formula as w ell. 44 • If φ and ψ are MSO-formulae, then φ ∧ ψ , φ ∨ ψ , φ → ψ , and φ ↔ ψ are MSO-form ulae as well. • If φ is an MSO-form ula, x is an individual v ariable, and X is a set v ariable, then ∃ xφ , ∀ xφ , ∃ X φ , and ∀ X φ are MSO-form ulae as well. Although “ → ” and “ ↔ ” are not explicitly necessary we list them for sak e of com- pleteness. Their semantics is analogous to ﬁrst-order logic by combining “ ∧ ”, “ ∨ ”, and “ ¬ ”. The constructions ha ve the following semantics: construct seman tics ¬ φ ( G, α ) | = ¬ φ ⇔ ( G, α ) 6| = φ φ ∧ ψ ( G, α ) | = φ ∧ ψ ⇔ ( G, α ) | = φ and ( G, α ) | = ψ φ ∨ ψ ( G, α ) | = φ ∨ ψ ⇔ ( G, α ) | = φ or ( G, α ) | = ψ ∃ x ( G, α ) | = ∃ xφ ⇔ there exists an a ∈ U suc h that ( G, α a x ) | = φ ∀ x ( G, α ) | = ∀ xφ ⇔ for all a ∈ U it holds that ( G, α a x ) | = φ ∃ X ( G, α ) | = ∃ X φ ⇔ there exists an A ⊆ U suc h that ( G, α A X ) | = φ ∀ X ( G, α ) | = ∀ X φ ⇔ for all A ⊆ U it holds that ( G, α A X ) | = φ Herein α ξ δ denotes an assignmen t with α ξ δ ( δ ) = ξ and α ξ δ ( ζ ) = α ( ζ ) for all ζ 6 = δ . Analogously to ﬁrst order logic, an MSO- sentenc e is an MSO-form ula without free v ariables. Now, w e are ready to present the main result in this ﬁeld according to ﬁxed-parameter algorithms. T o this end, w e deﬁne the following problem: MSO-Check Given: A graph G and an MSO-sentence ϕ . Question: Is there an assignment α such that ( G, α ) | = ϕ ? It is easy to see that one can reduce every graph problem that is based on a graph prop ert y , expressible b y an MSO-sentence, to MSO-Check : Compute an MSO- sen tence that expresses the graph prop ert y and tak e the graph and the sen tence as input for the MSO-Check algorithm. Courcelle developed the following imp ortan t theorem [Cou90]: Theorem 6 (Courcelle’s Theorem) . MSO-Check is ﬁxe d-p ar ameter-tr actable with r esp e ct to the c ombine d p ar ameter (tw( G ) , | ϕ | ) . Mor e over, ther e is a c omputable function f and an algorithm that solves MSO-Check in time f (tw( G ) , | ϕ | ) · | G | + O ( | G | ) . W e end with some simple examples for the application of Theorem 6. Examples and extension. W e start with a simple and w ell-studied graph prop erty: 3-c olor ability . A (vertex-) c oloring of an undirected graph is a mapping from the set of v ertices to a (ﬁnite) set of colors, suc h that no t w o adjacen t v ertices ha ve the same color. W e sa y that the graph is 3-colorable if there is a coloring with three colors. The graph prop ert y “to b e 3-colorable” is expressible with an MSO-sen tence: ϕ = ∃ C 1 ∃ C 2 ∃ C 3  ∀ x : V x → ( C 1 x ∨ C 2 x ∨ C 3 x )  ∧  ∀ e, ∀ a 6 = b : ( E e ∧ I ae ∧ I be ) → ¬ (( C 1 a ∧ C 1 b ) ∨ ( C 2 a ∧ C 2 b ) ∨ ( C 3 a ∧ C 3 b ))  . 5 Minimum Degree Deletion 45 Th us, due to Theorem 6 to decide whether a graph is 3-colorable is ﬁxed-parameter tractable with resp ect to the parameter treewidth. Arn b org et al. [ALS91] generalized this theorem to the case of “extended monadic second-order logic”. Here, w e can additionally mak e use of set cardinalities. This also includes the optimization problems regarding the set sizes. In this work, we only need the op eration “min X : P ( X )” that expresses that a sp eciﬁc predicate P holds for a minimum-size v ertex set X . Indep enden tly , Borie et al. [BPT92] obtained similar results, but from a more algorithmic p oin t of view. In their Theorem 3.5, they explicitly sa y that if a “minim um edge/v ertex deletion problem“ Prob that is based on a graph prop erty which is expressible in MSO-logic, then Prob is expressible in extended monadic second-order logic. T o b ecome familiar with the minimization extension, w e giv e an easy example here: Let G = ( V , E ) b e an undirected graph. A vertex c over is a subset C of vertices in V such that ev ery edge in E is incident to at least one vertex in C . The predicate VC( X ) :=“to b e a vertex cov er”, with X b eing a v ertex set, is expressible in MSO-logic: V C( C ) = ∀ e : E e → ( ∃ v : ( C v ∧ I v e )) . The optimization v arian t of vertex co v er where one asks for a minimum-size vertex co ver is expressible in extended MSO-logic: min C : VC( C ) . W e use extended MSO in the next section to in v estigate the parameterized com- plexit y of Minimum Degree Deletion with resp ect to the parameter treewidth. 5.3.2 MSO exp ression for Minimum Degree Deletion In the following, we giv e a monadic second order sen tence to prov e the ﬁxed- parameter tractability for Minimum Degree Deletion with resp ect to the pa- rameter t w :=“treewidth of the input graph”. Since s v ≥ t w it follows that Minimum Degree Deletion is also ﬁxed-parameter tractable with resp ect to the parameter s v . W e start with an observ ation that helps us gaining an alternativ e view on the prob- lem. Observ ation 9. L et G = ( V , E ) b e an undir e cte d gr aph with tr e ewidth t w . L et M ∗ b e any solution set. L et C b e a tr e e de c omp osition of G − M ∗ with maximum b ag-size t w + 1 and C is minimal, that is, it is not p ossible is obtain another tr e e de c omp osition by r emoving vertic es fr om the b ags. It holds that w c has de gr e e of at most t w − 1 in G − M ∗ . Pr o of. Assume tow ards a contradiction that w c has degree at least t w in G − M ∗ . Let L b e a leaf bag of C . There is a vertex v l that is only in bag L , b ecause C is minimal. Since L has size t w + 1 by deﬁnition, v l can hav e at most t w neigh b ors; a con tradiction to the fact that w c is the only v ertex of minimum degree in G − M ∗ . Inspired by Observ ation 9 we can formalize Minimum Degree Deletion as fol- lo ws: 46 Minimum Degree Deletion ∗ Given: An undirected graph G = ( V , E ), a distinguished vertex w c ∈ V , an in teger k ≥ 1, and an integer i ≤ t w . Question: Is there a subset K ⊆ V \ w c of size k such that w c has exactly i neigh b ors not in K and eac h other vertex is in K or has more than i neigh b ors not in K . It is easy to see that ( G, k ) is a yes-instance of Minimum Degree Deletion if and only if ( G, k , i ) is a yes-instance of Minimum Degree Deletion ∗ for some i ≤ t w . Hence, we describ e an MSO-sentence ϕ expressing the graph prop erty given b y the question of Minimum Degree Deletion ∗ . F or the ease of presentation w e ﬁrst describ e some of the parts of the sentence: The formula part adj( x, y ) is satisﬁable if and only if x and y are adjacent: adj( x, y ) = V x ∧ V y ∧ ( ∃ e ( E e ∧ I xe ∧ I y e )) . The formula part iNotKNeigh b ors( x, K , i ) is satisﬁable if and only if there are at least i neigh b ors of x which are not contained in the vertex subset K : iNotKNeigh b ors( x, K , i ) = ∃ n 1 ∃ n 2 . . . ∃ n i  ^ 1 ≤ a ≤ i adj( x, n a ) ∧ ¬ K n a  ∧  ^ 1 ≤ a d − k ; a conﬂict. Assume that w c has degree less than k in G − M d . In this case, M d m ust contain at least d − ( k − 1) neighbors of w c ; a conﬂict. Observ ation 13. The gr aph G has a vertex c over of size k + 1 + d which do es not c ontain w c . Pr o of. The set C ∪ U is a vertex cov er: G − ( C ∪ U ) do es not contain any edge; C is of size k + 1 and U of size d . Lemma 10. If ( G, w c , k 0 ) is a yes-instanc e of Minimum Degree Deletion then ( U ∗ , S ∗ , k ) is a yes-instanc e of Small Universe Hitting Set . 5 Minimum Degree Deletion 61 Pr o of. It remains to show that there is a hitting set U 0 ⊆ U ∗ . One can build U 0 as follows: F or each v ertex u i ∈ U whic h is not in M d add the element u ∗ i to U 0 . Due to Observ ation 12 the size of U 0 is k . It remains show that U 0 is a hitting set. Assume that there is a subset s ∗ j with j ∈ { 1 , . . . , n } that has no intersection with an y element in U 0 . Th us, for eac h element u ∗ i ∈ s ∗ j the corresp onding vertex u i is in M d . Due to the construction of G the vertex s j has degree k in G − M d . It follo ws that M d is not a solution set; a conﬂict. Lemma 11. Ther e is a p olynomial time and p ar ameter tr ansformation fr om Small Universe Hitting Set with r esp e ct to the c ombine d p ar ameter ( d, k ) to Minimum Degree Deletion with r esp e ct to the c ombine d p ar ameter ( s ∗ c , k 0 ) . Pr o of. Due to Lemma 9 and 10 the equiv alence of b oth instances (p oin t 1 of Deﬁni- tion 14) is given. Due to Observ ation 13 and the construction of graph G , the new parameter ( s ∗ c , k 0 ) is b ounded by a p olynomial only dep ending on the old parameter ( d, k ) (p oin t 2 of Deﬁnition 14): ( s ∗ c , k 0 ) = ( d + 1 + k , d − k ) . Putting all together, w e arrive at the following theorem: Theorem 12. Minimum Degree Deletion has no p olynomial kernel with r esp e ct to the c ombine d p ar ameter ( s ∗ c , k ) , with s ∗ c b eing the size of a vertex c over that do es not c ontain the distinguishe d vertex and k b eing the numb er of to delete d vertic es, unless coNP ⊆ NP / p oly . Of course, this implies that there is no hop e for p olynomial k ernels for the parameters t w , s v , s ∗ v and s c :=“size of a vertex cov er” as single parameters, or combined with k . 6 Bounded Degree Deletion In this c hapter, we analyze the parameterized complexity of Bounded Degree Deletion . The problem is motiv ated as graph problem where one searches for a v ertex subset of size at most k whose remo v al from the graph is a graph in whic h eac h v ertex has degree at most d . The follo wing section summarizes kno wn results of the parameterized complexity of Bounded Degree Deletion whic h are obtained from [Mos10]. 6.1 Kno wn results Bounded Degree Deletion is ﬁxed-parameter tractable with resp ect to the parameter k for constan t d , which can b e seen b y reduction to ( d + 2) -Hitting Set . F or d = 1 there is a 15 k -vertex k ernel and a O (2 k · k 2 · k n ) algorithm. An O ( k 1+  )-v ertex kernel w as show for d ≤ 2. F urthermore, there is a ﬁxed-parameter algorithm with running time O (( d + 2) k + n ( k + d )). F or un b ounded d , Bounded Degree Deletion is W[2]-complete for the parameter k . In the following w e start with inv estigating the parameterized complexit y with resp ect to the parameters that measure the “degree of acyclicity” b eginning with the “size of a feedback edge set”. The parameterized complexit y for “treewidth of the input graph” or “size of a feedbac k vertex set” remains op en in this work. 6.2 Size of a feedback edge set as pa rameter In this section, w e show ﬁxed-parameter tractability for Bounded Degree Dele- tion with resp ect to the parameter s e :=“size of a feedback edge set”. A ﬁrst step will b e to show that an annotated v ersion is p olynomial-time solv able on acyclic graphs. An annotated version on acyclic graphs. In the follo wing, w e suppose that the input graph is acyclic. W e describ e an algorithm that computes an optimal solu- tion set , that is, a v ertex subset of minimum size whose remo v al from the graph is a graph in which each v ertex has degree at most d . T o ﬁnally solv e Bounded Degree Deletion on graphs with b ounded feedbac k edge set size, we in tro duce a slightly mo diﬁed v ersion of Bounded Degree Deletion and sho w that it is solv able in p olynomial time on acyclic graphs. The mo diﬁed problem is deﬁned as follo ws: 64 Annot a ted Bounded Degree Deletion Given: An undirected graph G = ( V , E ), a vertex subset U , and integers d ≥ 0 and k ≥ 0. Question: Do es there exists a subset V 0 ⊆ ( V \ U ) of size at most k whose remo v al from G yields a graph in whic h eac h v ertex has degree at most d ? The vertex subset U is called the set of unr emovable vertices. The algorithm uses a sp ecialized b ottom-up tree-trav ersal and handles eac h vertex exactly once: Either the vertex will b e mark ed to b e “not con tained in the solution set” or the vertex will b e remov ed. This pro cess is called de cision step of the algorithm. The order of pro cessing the vertices corresp onds to their depth 1 (from higher to low er). V ertices that ha v e the same depth are handled in order of their degree (from higher to low er). This ensures three in v ariants in the decision step for v ertex x : 1. Each c hild of x w as either remov ed or was mark ed. 2. Each c hild of ev ery sibling of x was either remov ed or was mark ed. 3. There is no sibling of x with higher degree which w as not already remov ed or mark ed. The decision step of the algorithm is given in Figure 6.1. It is easy to see that after eac h decision step for a vertex x either: • x was remov ed, or • x was marked and has degree at most d , or • the algorithm canceled due to the detection of a no-instance. Let M b e the set of mark ed vertices and S := V \ M . Lemma 12. The algorithm c omputes an optimal solution set in O ( n 2 ) time. Pr o of. It remains to show that if the decision step was correct and optimal for each c hild of a vertex x , then the decision step is also correct and optimal for x . Due to the pro cessing order, the correctness (and optimality) of the whole algorithm follo ws. This pro of is more or less a complete induction: The base clause is “the decision step is correct and optimal for eac h child of a leaf ” whic h is clearly given, b ecause a leaf has no child. In the following we show correctness and optimality for 1 The depth of a vertex x in a tree is the length of the path b etw een x and the root. 6 Bounded Degree Deletion 65 Decision step Input: An undirected acyclic graph G and a v ertex x from G . Require: Each child of x w as either remov ed or was marked. Eac h c hild of ev ery sibling of x was either remov ed or was mark ed. There is no sibling of x with higher degree whic h was not already remov ed or mark ed. Let p denote the paren t vertex of x and let p p denote the paren t of p . Case A x is unremov able Case A.1 If deg( x ) = d + 1 and p is remov able, then remov e p and mark x to b e “not con tained in the s olution set”. Case A.2 If deg( x ) = d + 1 and p is unremo v able, then cancel and return “no”. Case A.3 If deg ( x ) > d + 1, then cancel and return “no”. Case A.4 If deg ( x ) < d + 1, then mark x to b e “not con tained in the solution set”. Case B x is remov able Case B.1 p is remov able Case B.1.a If deg ( x ) < d + 1, then mark x to b e “not contained in the solution set”. Case B.1.b If deg ( x ) > d + 1, then remov e x . Case B.1.c If deg( x ) = d + 1, then remov e the parent v ertex of x . If there is no paren t vertex ( x is the ro ot), then just remov e x . Case B.2 p is unremov able Case B.2.a If deg ( x ) ≥ d + 1, then remov e x . Case B.2.b If deg ( x ) < d + 1 and deg ( p ) < d + 1, then mark x to b e “not con tained in the solution set”. Case B.2.c It holds that deg ( x ) < d + 1 and deg ( p ) ≥ d + 1. If p p exists and p p is remo v able, then remov e p p . If deg ( p ) ≥ d + 1 (p ossibly after remo ving p p ), then remo ve x . Ensure: Either x is remov ed, or x is mark ed and has degree at most d , or the algorithm cancels due to the detection of a no-instance. F urthermore, the decision is optimal with resp ect to the total n um b er of remov ed vertices. Figure 6.1: Decision step of the algorithm. The paren t vertex of x is denoted as p . 66 eac h case: Case A.1 Since the decision w as correct for each child, p is the only neighbor of x that can b e remo ved. Remo ving x is not allow ed due to the problem deﬁnition. T o decrease the degree of x to d one must remo ve p . Case A.2 No neigh b or of x can b e remov ed. Hence, it is not p ossible to decrease the degree of x . Case A.3 Analogously to case A.1, p is the only neigh b or of x that can b e remo ved. Removing x is not allow ed due to the problem deﬁnition. Th us, it is not p ossible to decrease the degree of x b y more than one an ywa y . Case A.4 This case is correct by the problem deﬁnition. Case B.1.a Remo ving x do es not lead to an optimal solution. The degree of x is already small enough and p is remov able. Hence, it is at least as go o d to remo v e p instead of x . Case B.1.b Clearly , one must remov e either x or at least 2 neigh b ors of x . Since the decision was correct for each child, p is the only neighbor of x that can b e remo ved. Hence, one must remov e x an ywa y . Case B.1.c Clearly , one must remo v e either x or p . It is not necessary to remo ve x , b ecause remo ving p instead is alw ays b etter. Note that the degree of eac h child of x is already at most d . Case B.2.a No neighbor of x can b e remov ed. One must decrease the degree of x at least b y one. Clearly , the only wa y to do this is removing x . Case B.2.b Remo ving x do es not lead to an optimal solution. The degree of x and the degree of the only neigh b or p is already small enough so that w e do not need to remo ve x . Hence, not remo ving x cannot b e wrong. Case B.2.c One m ust remo v e enough neighbors of p suc h that it has ﬁnal degree at most d . Due to the inv ariants, no remaining sibling has another neigh b or than p and no sibling hat degree more than d . M ore precisely , x has degree at most d and siblings with higher degree w ere pro cessed b efore. It do es not matter which c hild of p will b e remo ved, but removing the paren t vertex can p ossibly decrease the degree of another v ertex. Hence, the v ertex subset S is an optimal solution set. It remains to prov e the running time. Collecting the degree information for each v ertex and ordering the v ertices according to their depth and degree tak es O ( n 2 ) time. Eac h decision step is computable in O ( n ) time: Marking a v ertex or chec king whether a vertex is marked tak es constan t time. Removing a vertex while up dating the degree information and up dating the ordering of the v ertices tak es O ( n ) time. Since eac h vertex is only pro cessed once, the algorithm needs n decision steps. Thus, the o v erall running time is in O ( n 2 ). Extension to the case of b ounded feedback edge set size. Let ( G, d, k ) b e an instance of Bounded Degree Deletion . In the following, w e assume that there is a feedbac k edge set E f of size s e . The ﬁrst step is a search tree that transforms 6 Bounded Degree Deletion 67 the instance in to acyclic instances. W e branch on the feedback edge set elemen ts in to three cases. Let { x, y } b e an edge in E f : Branching case 1 Remov e x . Branching case 2 Remov e y . Branching case 3 Do not remov e x and y . Instead, remov e { x, y } and add one additional leaf a x to x and one additional leaf a y to y . (This is necessary to preserv e the degrees.) Mark x , a x , y , and a y as “unremo v able”. Let G 0 b e the resulting graph, U b e the set of v ertices that are mark ed as “un- remo v able”, and r the num b er of remo ved v ertices. In eac h search tree leaf G 0 is acyclic. W e ha ve to guarantee to ﬁnd an optimal solution set in at least one branch. Th us, a second step is to solve the Annot a ted Bounded Degree Deletion instance ( G 0 , U, d, k − r ) in each search tree leaf. If the Annot a ted Bounded Degree Deletion instance in an y searc h tree leaf is a y es-instance, then return “y es”. Otherwise, return “no”. Putting all together, we arrive at the following: Theorem 13. Bounded Degree Deletion c an b e solve d in O (3 s e · n 2 ) time with s e b eing the size of a fe e db ack e dge set. Pr o of. The running time of the algorithm is clearly in O (3 s e · n 2 ). F urthermore, it is easy to see that if there is a y es-instance of Annot a ted Bounded Degree Deletion in a search tree leaf, then the original Bounded Degree Deletion instance is a yes-instance, to o. It remains to show that if the original instance is a yes-instance of Bounded Degree Deletion , then there is a yes-instance of Annot a ted Bounded Degree Deletion in at least one search tree leaf. Assume, that there is a solution set M d for the original instance ( G, d, k ). Let V I denote the vertices that are incident to an edge from E f . Let M I := V I ∩ M d denote the solution set vertices that are inciden t to an edge from E f . Let M 2 I := { x | x ∈ M I ∧ ∃ y : y ∈ M I ∧ { x, y } ∈ E f } denote the solution set vertices that are connected to another solution set vertex b y an edge from E f . It is easy to verify that there is a search tree leaf s with U s b eing the set of as “unremov able” marked v ertices and R s is the set of remo ved vertices such that: • F or each vertex x ∈ M 2 I either x ∈ R s and y / ∈ U s or y ∈ R s and x / ∈ U s . • F or each vertex x ∈ ( M I \ M 2 I ) it holds that x ∈ R s . • Each v ertex u ∈ U s is not part of the solution set M d . Clearly , the Annot a ted Bounded Degree Deletion instance in the searc h tree leaf s is a y es-instance. 7 Conclusion and Outlo ok W e inv estigated the parameterized complexity of three similar vertex deletion prob- lems. Since each problem is solv able in p olynomial time when restricted to acyclic graphs, but NP-hard in general, it was natural to study ﬁxed-parameter tractability with resp ect to parameters that measure the “degree of acyclicity”. More precisely , w e considered s e :=“size of a feedbac k edge set” resp ectiv ely s a :=“size of a feed- bac k arc set”, s v :=“size of a feedback vertex set”, and t w :=“treewidth of the input graph”. F or Minimum Indegree Deletion , which is equiv alen t to constructive con trol by deleting candidates in Llull elections, we show ed that it is W[2]-hard with resp ect to the parameters s a and s v . In addition, it is at least in XP with resp ect to s v (and s a ). Although it is W[2]-complete with resp ect to the parameter k :=“solution size”, w e could show ﬁxed-parameter tractabilit y with resp ect the com bined pa- rameters ( s a , k ) and ( s v , k ). The ﬁxed-parameter algorithm in Section 4.3 solv es Minimum Indegree Deletion in O ( s v · ( k + 1) s v · n 2 ) time. There should b e ro om for impro vemen ts of this algorithm. In contrast to Minimum Indegree Deletion , Minimum Degree Deletion is ev en ﬁxed-parameter tractable with resp ect to each of the parameters that measure the “degree of acyclicit y”. W e sho wed that Minimum Degree Deletion has no p olynomial k ernel with respect to s v or t w , unless the p olynomial-time hierarc h y collapses. Moreo ver, there is also no p olynomial kernel with resp ect to the com- bined parameter ( s ∗ c , k ) with s ∗ c :=“size of a vertex cov er that do es not contain w c ”. Ho wev er, a v ertex-linear kernel w as found with resp ect to s e . In future researc h it w ould b e desirable to develop a practicable algorithm with resp ect to t w . 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Graph and Election Problems Parameterized by Feedback Set Numbers

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